Quantum Protocol Zoo - User contributions [en]

Quantum Gate Set Tomography

2020-06-18T09:30:03Z

Rhea: /* Outline */

Quantum gate set tomography is a robust and powerful method for self-consistently characterizing an entire set of quantum logic gates on a black-box quantum device. This method arose from the observation that [[Full Quantum Process Tomography with Linear inversion]] is inaccurate in the presence of state preparation and measurement errors. This procedure estimates the complete process matrices of Pauli transfer matrices for the gate set based on experimental data.

'''Tags:''' [[Certification protocol]], [[Tomography]], [[Quantum process density matrix reconstruction]], [[Maximum Likelihood estimation]], [[Linear Inversion]], [[Gate set]]

==Assumptions==

* The selected measurement basis should be tomographically complete.
* For the calculation of the likelihood function we assume that the noise on the coincidence measurements has a Gaussian probability distribution. We also assume that each of our measurements is taken for the same amount of time.
* The SPAM gates are composed of gates in the gate set and therefore the minimal gate set must include sufficient gates to create a complete set of states and measurements.
* In the gate set, the first gate is always selected to be the null gate.
* We assume no knowledge about the gate set.

==Outline==

[[State Tomography]] and [[Process Tomography]] assume that the initial states and measurements are known. But if the [[state preparation and measurement]] (SPAM) gates are faulty then the estimates provided using these techniques are faulty as well. Quantum gate set tomography solves this problem by including the SPAM gates self-consistently in the gate set to be estimated.

The goal of this method is to completely characterize a gate set, which includes an unknown set of gates and an initial state and 2-outcome [[POVM]]. For self-consistency, the SPAM gates are treated on the same footing as the original gates. The SPAM gates are composed of gates in the gate set and therefore the minimal gate set must include sufficient gates to create a complete set of states and measurements. When these gates are applied to our unknown fixed state and measurement, a complete set of initial states and final state is produced. The experimental requirement for GST is the ability to measure the expectation values for the gate set.

Two techniques are shown here, which are Linear inversion (LGST) and maximum likelihood estimation (MLE). The linear inversion protocol is fairly easy to implement numerically and is useful for providing quick diagnostics without resorting to more computationally intensive estimation via constrained optimization. The LGST estimate is not generally physical, the information obtained is somewhat qualitative, and hence MLE is preferred. LGST is a useful starting point for MLE. Since the starting point must be physical, while LGST is not, we should use the closest physical gate set to the LGST estimate.

This method consists of the following steps:

* Initialize the qubit to a particular state. In most systems, the natural choice for this state is the ground state of the qubit.
* For a particular choice of SPAM gates and a gate from the Gate set, the corresponding gate sequence is applied to the qubit. As the SPAM gates are composed of the gates from the gate set, the sequence only contains the gates from the gate set.
* The 2-outcome POVM is measured. This POVM is required to be a positive semi-definite Hermitian operator. The natural choice for this POVM in most systems is <math>|0\rangle\langle 0|</math>. (Sometimes <math>|1\rangle\langle 1|</math> is used)
* The above 1-3 steps are repeated a large number of times (10000 to 100000). For every repetition, the measurement is success (i.e., the measured state is <math>|0\rangle\langle 0|</math>) is recorded as 1 and the measurement failure (i.e., the measured state is not <math>|0\rangle\langle 0|</math>) is recorded as 0.
* The measurement results in the step above are averaged over the number of times the measurement was repeated and the expectation value is calculated.
* The above 1-5 steps are repeated for every SPAM gates and every gate in the gate set. This gives us the probability of all the possible gate set and SPAM gate combinations.
* Repeat steps 1-5 to measure the expectation values of only the SPAM gates. This is used to form a matrix known as the Gram matrix. If the first gate of the gate set is a null gate, then this step is not needed as this data already exists due to the step performed above.
* To find the gate set estimate from the measurement data, there are two techniques:
** Linear inversion method (LGST):
*** We should check that the gram matrix is invertible and after certain calculations with the inverted gram matrix, the gate set can be estimated. The LGST estimate is not generally physical, the information obtained is somewhat qualitative.
*** The gate set estimated in this way is in a different gauge from the actual gate set. To compensate, we transform to a more useful gauge. Since we do not know the actual gate set, only the target one, the most useful gauge is the one that brings the estimated gate set as close as possible based, on some distance metric, to the target. The gauge transformation is found by solving an optimization problem and the resulting gauge matrix is then applied to the gate set found in the step above.
** Maximum likelihood estimation (MLE): LGST is a useful starting point for MLE. The goal in this method is to find the true probabilities corresponding to the measurements, subject to physical constraints. In this method, the best estimate is found by fitting to experimental data a theoretical model of the probability of obtaining that data.
*** This method begins by parameterizing the estimate, state and measurement matrices in terms of a vector of parameters. A number of constraints reduce the total number of independent parameters.
*** Putting everything together, the probability estimates are written in terms of the parameter vector and Depending on the parameterization choice each of the gates and states is either a linear or quadratic function of its parameters. Each of the gates and states is either a linear or quadratic function of its parameters. The estimator is, therefore, a homogeneous function of order 5 (linear parameterization) or 10 (quadratic parameterization).
*** MLE proceeds by finding the set of parameters that minimizes an objective function. The objective, or likelihood function, is the probability distribution we assume produced the data.
*** There are two kinds of parameterizations for gates that are commonly used: The Pauli Process Matrix representation and The Pauli transfer Matrix Representation.

==Hardware Requirements==

* Measurement device.

==Notation==

* <math>d</math>: Dimension of Hilbert space
* <math>G</math>: Gate set. <math>G = \{G_0, G_1, .. G_K\}</math>. Generally <math>G_0 = {}</math> null gate.
* <math>F</math>: SPAM gate set. <math>F = \{F_0, ... , F_N\}</math>. <math>F</math> is composed of gates in <math>G</math>.
* <math>|\rho\rangle</math>: Initial quantum state. Generally the natural choice for <math>\rho</math> is <math>|0\rangle\langle 0|</math>
* <math>E</math>: 2-outcome POVM.
* <math>n</math>: Number of times the measurement is repeated.
* <math>n_r</math>: Result of <math>r^{th}</math> measurement.
* <math>m_{ijk}</math>: measurement of the expectation value <math>p_{ijk}</math>
* <math>p_{ijk}</math>: Expectation value. <math>p_{ijk} = \langle E|F_iG_kF_j|\rho\rangle</math>
* <math>g_{ij}</math>: Gram matrix
* <math>B_{ij}</math>: Gauge matrix
* <math>\vec{t}</math>: Parameter vector
* <math>\hat{G}(\vec{t})</math>: Parameterised estimate matrix. This has <math>d^4</math> parameters.
* <math>|\hat{p}(\vec{t})\rangle</math>: Parameterised state matrix. This has <math>d^2</math> parameters.
* <math>\langle \hat{E}(\vec{t})|</math>: Parameterised measurement matrix. This has <math>d^2</math> parameters.
* <math>l(\hat{G})</math>: likelihood function.
* <math>\sigma_{ijk}</math>: The sampling variance in the measurement <math>m</math>. <math>\sigma^2 = p(1-p)/n</math>
* <math>\chi_G</math>: Process matrix for gate <math>G</math> is defined in terms of the gate’s action on an arbitrary state <math>\rho</math> to produce a new state <math>G(p)</math>. <math>G(p) = \sum_{i,j=1}^{d^2}(\chi_G)_{ij}P_i\rho P_j</math>. <math>\chi</math> is a Hermitian positive semidefinite matrix, written in terms of <math>\chi = T^{\dagger}T</math>
* <math>T</math>: a lower-diagonal complex matrix, <math>
\hat{T}(t) = \begin{bmatrix}
t_1 & 0 & ... & 0 \\
t_{2^n + 1} + it_{2^n+2} & t_2 & ... & 0 \\
... & ... & ... & 0 \\
t_{4^n -1} + it_{4^n} & t_{4^n -3} + it_{4^n - 2} & t_{4^n - 5} + it_{4^n - 4} & t_{2^n}
\end{bmatrix}
</math>
* <math>P_i</math>: Pauli operator <math>i</math> acting on the state
* <math>R_G</math>: Pauli tranfer matrix for a gate G is defined in terms of the gate’s action on Pauli matrices. <math>(R_G)_{ij} = Tr\{P_iG(P_j)\}. |G(\rho)\rangle = R_G|\rho\rangle</math>

==Properties==

* Figure of merit: Gate set
* Multiple copies of the quantum state are required in this method.
* <math>E_j</math> should be tomographically complete.
* In contrast to QPT, it is not possible in GST to characterize one gate at a time. Instead, GST estimates every gate in the gate set simultaneously.
* SPAM gates are included in the gate set for self-consistency.
* The limitation of linear-inversion GST (or LGST) is that it does not constrain the estimates to be physical. However, LGST provides a convenient method for diagnosing gate errors and also gives a good starting point for the constrained maximum likelihood estimation.
* In the gate set, the first gate is generally selected to be the null gate.
* The simulated errors are of three different types, representing coherent and incoherent gate errors as well as intrinsic SPAM errors.

==Procedure Description==

'''Output''': Gate set, <math>G</math>

* Initialize system to <math>|\rho\rangle</math>
* For <math>i = 1, 2, ..., N</math>:
** For <math>j = 1, 2, ..., N</math>:
*** For <math>k = 1, 2, ..., K</math>:
**** For <math>r = 1, 2, ..., n</math>
***** Apply gate sequence <math>F_i \circ G_k \circ F_j</math>
***** Measure with POVM <math>E</math>, get <math>n_r = 1</math> or <math>n_r = 0</math>
**** <math>m_{ijk} = \sum_{r=1}^{n} \frac{n_r}{n}</math>
**** <math>(\tilde{G}_k)_{ij} = p_{ijk} = m_{ijk} = \langle E|F_iG_kF_j|\rho\rangle</math>
**** if <math>k==0</math> (null gate), <math>g_{ij} = \langle E|F_iF_j|\rho\rangle</math>
** <math>|\tilde{\rho}\rangle_i = \langle E|F_i|\rho\rangle</math>
** <math>B_{ij} = \langle i |F_j |\rho\rangle</math>
* For Linear inversion:
** Check that the Gram matrix <math>g_{ij}</math> is non singular, so that it may be inverted
** For the gate set, the estimate is <math>|\hat{\rho}\rangle = g^{-1}|\tilde{\rho}\rangle, \langle \hat{E}| = \langle \tilde{E} |, \hat{G_k} = g^{-1} \tilde{G}_k</math>
** Apply gauge optimization, by minimizing:
<div style="text-align: center;"><math>\hat{B}^* = argmin_{\hat{B}}\sum^{K+1}_{k=1}Tr\{(\hat{G}_k - \hat{B}^{-1}T_k\hat{B})^T(\hat{G}_k - \hat{B}^{-1}T_k\hat{B})\}
</math></div>
** Final gate set is: <math>\hat{G}_k^{*} = \hat{B}^*\hat{G}_k(\hat{B}^*)^{-1}, |\hat{p}^*\rangle = \hat{B}^*|\hat{p}\rangle, \langle \hat{E}^*| = \langle \hat{E}|(\hat{B}^*)^{-1}</math>
* For Maximum Likelihood Estimation:
** <math>\hat{p}_{ijk} = \langle \hat{E}(\vec{t})|\hat{F}_i(\vec{t})\hat{G}_k(\vec{t})\hat{F}_j(\vec{t})|\hat{\rho}(\vec{t})\rangle</math>
** Find the set of parameters <math>\vec{t}</math> where <math>l(\hat{G})</math> is minimized.
<div style="text-align: center;">Minimize: <math>
l(\hat{G}) = \sum_{ijk} (m_{ijk} - \hat{p}_{ijk}(\vec{t}))^2 / \sigma^2_{ijk}
</math></div>
** Minimize the above function with two different commonly used types of parameterisation of gates:
*** Pauli Process Matrix Optimization problem:
<div style="text-align: center;">Minimize: <math>
l(\hat{G}) = \sum_{ijk} (m_{ijk} - \sum_{mnrstu}(\chi_{F_i})_{tu} (\chi_{G_k})_{rs} (\chi_{F_j})_{mn} Tr\{EP_t P_r P_m \rho P_n P_s P_u\} )^2 / \sigma^2_{ijk}
</math></div>
<div style="text-align: center;">Subject to: <math>
\sum_{mn}(\chi_G)_{mn}Tr\{P_mP_rP_n\} - \delta_{0r} = 0, (r = 1, .. , d^2), \forall G \in Gateset
</math></div>
<div style="text-align: center;"><math>
Tr\{\rho\} = 1
</math></div>
<div style="text-align: center;"><math>
1 - E \geqslant 0
</math></div>
*** Pauli Transfer Matrix Representation
<div style="text-align: center;"> Minimize: <math>
l(\hat{G}) = \sum_{ijk} (m_{ijk} - \langle \hat{E}(\vec{t})|\hat{R}_{F_i}(\vec{t})\hat{R}_{G_k}(\vec{t})\hat{R}_{F_j}(\vec{t})|\hat{\rho}(\vec{t})\rangle )^2 / \sigma^2_{ijk}
</math></div>
<div style="text-align: center;">Subject to: <math>
\rho_G = \frac{1}{d^2} \sum_{i,j=1}^{d^2}(R_G)_{ij}P^T_G \otimes P_i \geqslant 0, \forall G \in Gateset
</math></div>
<div style="text-align: center;"><math>
(R_G)_{0i} = \delta_{0i}, \forall G \in Gateset
</math></div>
<div style="text-align: center;"><math>
(R_G)_{ij} \in [-1, 1], \forall G, i, j
</math></div>
<div style="text-align: center;"><math>
Tr\{\rho\} = 1
</math></div>
<div style="text-align: center;"><math>
1 - E \geqslant 0
</math></div>

==Further Information==

==Related Papers==
* Daniel Greenbaum (2015) arXiv:1509.02921v1: Introduction to Quantum Gate Set Tomography

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Quantum Gate Set Tomography

2020-06-18T09:29:14Z

Rhea:

Quantum gate set tomography is a robust and powerful method for self-consistently characterizing an entire set of quantum logic gates on a black-box quantum device. This method arose from the observation that [[Full Quantum Process Tomography with Linear inversion]] is inaccurate in the presence of state preparation and measurement errors. This procedure estimates the complete process matrices of Pauli transfer matrices for the gate set based on experimental data.

'''Tags:''' [[Certification protocol]], [[Tomography]], [[Quantum process density matrix reconstruction]], [[Maximum Likelihood estimation]], [[Linear Inversion]], [[Gate set]]

==Assumptions==

* The selected measurement basis should be tomographically complete.
* For the calculation of the likelihood function we assume that the noise on the coincidence measurements has a Gaussian probability distribution. We also assume that each of our measurements is taken for the same amount of time.
* The SPAM gates are composed of gates in the gate set and therefore the minimal gate set must include sufficient gates to create a complete set of states and measurements.
* In the gate set, the first gate is always selected to be the null gate.
* We assume no knowledge about the gate set.

==Outline==

[[Quantum state tomography]] and [[Quantum process tomography]] assume that the initial states and measurements are known. But if the [[state preparation and measurement]] (SPAM) gates are faulty then the estimates provided using these techniques are faulty as well. Quantum gate set tomography solves this problem by including the SPAM gates self-consistently in the gate set to be estimated.

The goal of this method is to completely characterize a gate set, which includes an unknown set of gates and an initial state and 2-outcome [[POVM]]. For self-consistency, the SPAM gates are treated on the same footing as the original gates. The SPAM gates are composed of gates in the gate set and therefore the minimal gate set must include sufficient gates to create a complete set of states and measurements. When these gates are applied to our unknown fixed state and measurement, a complete set of initial states and final state is produced. The experimental requirement for GST is the ability to measure the expectation values for the gate set.

Two techniques are shown here, which are Linear inversion (LGST) and maximum likelihood estimation (MLE). The linear inversion protocol is fairly easy to implement numerically and is useful for providing quick diagnostics without resorting to more computationally intensive estimation via constrained optimization. The LGST estimate is not generally physical, the information obtained is somewhat qualitative, and hence MLE is preferred. LGST is a useful starting point for MLE. Since the starting point must be physical, while LGST is not, we should use the closest physical gate set to the LGST estimate.

This method consists of the following steps:

* Initialize the qubit to a particular state. In most systems, the natural choice for this state is the ground state of the qubit.
* For a particular choice of SPAM gates and a gate from the Gate set, the corresponding gate sequence is applied to the qubit. As the SPAM gates are composed of the gates from the gate set, the sequence only contains the gates from the gate set.
* The 2-outcome POVM is measured. This POVM is required to be a positive semi-definite Hermitian operator. The natural choice for this POVM in most systems is <math>|0\rangle\langle 0|</math>. (Sometimes <math>|1\rangle\langle 1|</math> is used)
* The above 1-3 steps are repeated a large number of times (10000 to 100000). For every repetition, the measurement is success (i.e., the measured state is <math>|0\rangle\langle 0|</math>) is recorded as 1 and the measurement failure (i.e., the measured state is not <math>|0\rangle\langle 0|</math>) is recorded as 0.
* The measurement results in the step above are averaged over the number of times the measurement was repeated and the expectation value is calculated.
* The above 1-5 steps are repeated for every SPAM gates and every gate in the gate set. This gives us the probability of all the possible gate set and SPAM gate combinations.
* Repeat steps 1-5 to measure the expectation values of only the SPAM gates. This is used to form a matrix known as the Gram matrix. If the first gate of the gate set is a null gate, then this step is not needed as this data already exists due to the step performed above.
* To find the gate set estimate from the measurement data, there are two techniques:
** Linear inversion method (LGST):
*** We should check that the gram matrix is invertible and after certain calculations with the inverted gram matrix, the gate set can be estimated. The LGST estimate is not generally physical, the information obtained is somewhat qualitative.
*** The gate set estimated in this way is in a different gauge from the actual gate set. To compensate, we transform to a more useful gauge. Since we do not know the actual gate set, only the target one, the most useful gauge is the one that brings the estimated gate set as close as possible based, on some distance metric, to the target. The gauge transformation is found by solving an optimization problem and the resulting gauge matrix is then applied to the gate set found in the step above.
** Maximum likelihood estimation (MLE): LGST is a useful starting point for MLE. The goal in this method is to find the true probabilities corresponding to the measurements, subject to physical constraints. In this method, the best estimate is found by fitting to experimental data a theoretical model of the probability of obtaining that data.
*** This method begins by parameterizing the estimate, state and measurement matrices in terms of a vector of parameters. A number of constraints reduce the total number of independent parameters.
*** Putting everything together, the probability estimates are written in terms of the parameter vector and Depending on the parameterization choice each of the gates and states is either a linear or quadratic function of its parameters. Each of the gates and states is either a linear or quadratic function of its parameters. The estimator is, therefore, a homogeneous function of order 5 (linear parameterization) or 10 (quadratic parameterization).
*** MLE proceeds by finding the set of parameters that minimizes an objective function. The objective, or likelihood function, is the probability distribution we assume produced the data.
*** There are two kinds of parameterizations for gates that are commonly used: The Pauli Process Matrix representation and The Pauli transfer Matrix Representation.

==Hardware Requirements==

* Measurement device.

==Notation==

* <math>d</math>: Dimension of Hilbert space
* <math>G</math>: Gate set. <math>G = \{G_0, G_1, .. G_K\}</math>. Generally <math>G_0 = {}</math> null gate.
* <math>F</math>: SPAM gate set. <math>F = \{F_0, ... , F_N\}</math>. <math>F</math> is composed of gates in <math>G</math>.
* <math>|\rho\rangle</math>: Initial quantum state. Generally the natural choice for <math>\rho</math> is <math>|0\rangle\langle 0|</math>
* <math>E</math>: 2-outcome POVM.
* <math>n</math>: Number of times the measurement is repeated.
* <math>n_r</math>: Result of <math>r^{th}</math> measurement.
* <math>m_{ijk}</math>: measurement of the expectation value <math>p_{ijk}</math>
* <math>p_{ijk}</math>: Expectation value. <math>p_{ijk} = \langle E|F_iG_kF_j|\rho\rangle</math>
* <math>g_{ij}</math>: Gram matrix
* <math>B_{ij}</math>: Gauge matrix
* <math>\vec{t}</math>: Parameter vector
* <math>\hat{G}(\vec{t})</math>: Parameterised estimate matrix. This has <math>d^4</math> parameters.
* <math>|\hat{p}(\vec{t})\rangle</math>: Parameterised state matrix. This has <math>d^2</math> parameters.
* <math>\langle \hat{E}(\vec{t})|</math>: Parameterised measurement matrix. This has <math>d^2</math> parameters.
* <math>l(\hat{G})</math>: likelihood function.
* <math>\sigma_{ijk}</math>: The sampling variance in the measurement <math>m</math>. <math>\sigma^2 = p(1-p)/n</math>
* <math>\chi_G</math>: Process matrix for gate <math>G</math> is defined in terms of the gate’s action on an arbitrary state <math>\rho</math> to produce a new state <math>G(p)</math>. <math>G(p) = \sum_{i,j=1}^{d^2}(\chi_G)_{ij}P_i\rho P_j</math>. <math>\chi</math> is a Hermitian positive semidefinite matrix, written in terms of <math>\chi = T^{\dagger}T</math>
* <math>T</math>: a lower-diagonal complex matrix, <math>
\hat{T}(t) = \begin{bmatrix}
t_1 & 0 & ... & 0 \\
t_{2^n + 1} + it_{2^n+2} & t_2 & ... & 0 \\
... & ... & ... & 0 \\
t_{4^n -1} + it_{4^n} & t_{4^n -3} + it_{4^n - 2} & t_{4^n - 5} + it_{4^n - 4} & t_{2^n}
\end{bmatrix}
</math>
* <math>P_i</math>: Pauli operator <math>i</math> acting on the state
* <math>R_G</math>: Pauli tranfer matrix for a gate G is defined in terms of the gate’s action on Pauli matrices. <math>(R_G)_{ij} = Tr\{P_iG(P_j)\}. |G(\rho)\rangle = R_G|\rho\rangle</math>

==Properties==

* Figure of merit: Gate set
* Multiple copies of the quantum state are required in this method.
* <math>E_j</math> should be tomographically complete.
* In contrast to QPT, it is not possible in GST to characterize one gate at a time. Instead, GST estimates every gate in the gate set simultaneously.
* SPAM gates are included in the gate set for self-consistency.
* The limitation of linear-inversion GST (or LGST) is that it does not constrain the estimates to be physical. However, LGST provides a convenient method for diagnosing gate errors and also gives a good starting point for the constrained maximum likelihood estimation.
* In the gate set, the first gate is generally selected to be the null gate.
* The simulated errors are of three different types, representing coherent and incoherent gate errors as well as intrinsic SPAM errors.

==Procedure Description==

'''Output''': Gate set, <math>G</math>

* Initialize system to <math>|\rho\rangle</math>
* For <math>i = 1, 2, ..., N</math>:
** For <math>j = 1, 2, ..., N</math>:
*** For <math>k = 1, 2, ..., K</math>:
**** For <math>r = 1, 2, ..., n</math>
***** Apply gate sequence <math>F_i \circ G_k \circ F_j</math>
***** Measure with POVM <math>E</math>, get <math>n_r = 1</math> or <math>n_r = 0</math>
**** <math>m_{ijk} = \sum_{r=1}^{n} \frac{n_r}{n}</math>
**** <math>(\tilde{G}_k)_{ij} = p_{ijk} = m_{ijk} = \langle E|F_iG_kF_j|\rho\rangle</math>
**** if <math>k==0</math> (null gate), <math>g_{ij} = \langle E|F_iF_j|\rho\rangle</math>
** <math>|\tilde{\rho}\rangle_i = \langle E|F_i|\rho\rangle</math>
** <math>B_{ij} = \langle i |F_j |\rho\rangle</math>
* For Linear inversion:
** Check that the Gram matrix <math>g_{ij}</math> is non singular, so that it may be inverted
** For the gate set, the estimate is <math>|\hat{\rho}\rangle = g^{-1}|\tilde{\rho}\rangle, \langle \hat{E}| = \langle \tilde{E} |, \hat{G_k} = g^{-1} \tilde{G}_k</math>
** Apply gauge optimization, by minimizing:
<div style="text-align: center;"><math>\hat{B}^* = argmin_{\hat{B}}\sum^{K+1}_{k=1}Tr\{(\hat{G}_k - \hat{B}^{-1}T_k\hat{B})^T(\hat{G}_k - \hat{B}^{-1}T_k\hat{B})\}
</math></div>
** Final gate set is: <math>\hat{G}_k^{*} = \hat{B}^*\hat{G}_k(\hat{B}^*)^{-1}, |\hat{p}^*\rangle = \hat{B}^*|\hat{p}\rangle, \langle \hat{E}^*| = \langle \hat{E}|(\hat{B}^*)^{-1}</math>
* For Maximum Likelihood Estimation:
** <math>\hat{p}_{ijk} = \langle \hat{E}(\vec{t})|\hat{F}_i(\vec{t})\hat{G}_k(\vec{t})\hat{F}_j(\vec{t})|\hat{\rho}(\vec{t})\rangle</math>
** Find the set of parameters <math>\vec{t}</math> where <math>l(\hat{G})</math> is minimized.
<div style="text-align: center;">Minimize: <math>
l(\hat{G}) = \sum_{ijk} (m_{ijk} - \hat{p}_{ijk}(\vec{t}))^2 / \sigma^2_{ijk}
</math></div>
** Minimize the above function with two different commonly used types of parameterisation of gates:
*** Pauli Process Matrix Optimization problem:
<div style="text-align: center;">Minimize: <math>
l(\hat{G}) = \sum_{ijk} (m_{ijk} - \sum_{mnrstu}(\chi_{F_i})_{tu} (\chi_{G_k})_{rs} (\chi_{F_j})_{mn} Tr\{EP_t P_r P_m \rho P_n P_s P_u\} )^2 / \sigma^2_{ijk}
</math></div>
<div style="text-align: center;">Subject to: <math>
\sum_{mn}(\chi_G)_{mn}Tr\{P_mP_rP_n\} - \delta_{0r} = 0, (r = 1, .. , d^2), \forall G \in Gateset
</math></div>
<div style="text-align: center;"><math>
Tr\{\rho\} = 1
</math></div>
<div style="text-align: center;"><math>
1 - E \geqslant 0
</math></div>
*** Pauli Transfer Matrix Representation
<div style="text-align: center;"> Minimize: <math>
l(\hat{G}) = \sum_{ijk} (m_{ijk} - \langle \hat{E}(\vec{t})|\hat{R}_{F_i}(\vec{t})\hat{R}_{G_k}(\vec{t})\hat{R}_{F_j}(\vec{t})|\hat{\rho}(\vec{t})\rangle )^2 / \sigma^2_{ijk}
</math></div>
<div style="text-align: center;">Subject to: <math>
\rho_G = \frac{1}{d^2} \sum_{i,j=1}^{d^2}(R_G)_{ij}P^T_G \otimes P_i \geqslant 0, \forall G \in Gateset
</math></div>
<div style="text-align: center;"><math>
(R_G)_{0i} = \delta_{0i}, \forall G \in Gateset
</math></div>
<div style="text-align: center;"><math>
(R_G)_{ij} \in [-1, 1], \forall G, i, j
</math></div>
<div style="text-align: center;"><math>
Tr\{\rho\} = 1
</math></div>
<div style="text-align: center;"><math>
1 - E \geqslant 0
</math></div>

==Further Information==

==Related Papers==
* Daniel Greenbaum (2015) arXiv:1509.02921v1: Introduction to Quantum Gate Set Tomography

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Randomised Benchmarking

2020-05-31T08:04:05Z

Rhea: /* Functionality Description */

==Functionality Description==
Randomized benchmarking refers to a collection of methods that aim to reliably estimating the magnitude of an average error of a quantum gate set in a robust fashion against state preparation and measurement error. It achieves this goal by applying sequences of feasible quantum gates of varying length, so that small errors are amplified with the sequence length leading

From a pragmatic point of view, RB protocols thereby define benchmarks that can be used to compare different digital quantum devices.In important instances ([[Standard Randomised Benchmarking]]), the benchmark can be related to the average gate fidelity, rendering RB protocols flexible certification tools. To this end, a group structure of the gate set is made use to achieve two goals: On the one hand, this is to control the theoretical prediction of error-free sequences.

RB is most prominently considered for Clifford gates and has been extended to other finite groups. Assumptions on having identical noise levels per gate have been lessened ([[Standard Randomised Benchmarking]]) and [[Randomised Benchmarking with confidence]] introduced. RB schemes have been generalized to other measures of quality, such as relative average gate fidelities ([[Interleaved Randomised Benchmarking]]), the unitarity [[Purity Benchmarking]], measures for losses [[Robust characterization of loss rates]], leakage [[Robust characterization of leakage errors]], addressibility [[Characterization of addressability by randomized benchmarking]] or even tomographic schemes that combine data from multiple RB experiment.

==Protocols==
* [[Characterization of addressability by randomized benchmarking]]
* [[Interleaved Randomised Benchmarking]]
* [[Purity Benchmarking]]
* [[Randomised Benchmarking with confidence]]
* [[Robust characterization of leakage errors]]
* [[Robust characterization of loss rates]]
* [[Standard Randomised Benchmarking]]

==Properties==
* The figure of merits in different protocols are: average error rate, average fidelity of a noise quantum circuit, unitarity, measures for losses, leakage, addressibility and cross-talk.

==Related Papers==
* E.Knill et al (2007) arXiv:0707.0963: gate and time-independent noise model
* E. Mageson et al (2011) arXiv:1009.3639: multi-parameter model
* Magesan et al. PRL (2012): Interleaved Randomized Benchmarking
* Harper et al (2016) arXiv:1608.02943v2: Interleaved Randomised Benchmarking to estimate fidelity of T gates
* Wallman, Granade, Harper, F., NJP 2015: Purity benchmarking

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Randomised Benchmarking

2020-05-31T08:02:58Z

Rhea: /* Protocols */

==Functionality Description==
Randomized benchmarking refers to a collection of methods that aim to reliably estimating the magnitude of an average error of a quantum gate set in a robust fashion against state preparation and measurement error. It achieves this goal by applying sequences of feasible quantum gates of varying length, so that small errors are amplified with the sequence length leading

From a pragmatic point of view, RB protocols thereby define benchmarks that can be used to compare different digital quantum devices.In important instances ([[Standard Randomised Benchmarking]]), the benchmark can be related to the average gate fidelity, rendering RB protocols flexible certification tools. To this end, a group structure of the gate set is made use to achieve two goals: On the one hand, this is to control the theoretical prediction of error-free sequences.

RB is most prominently considered for Clifford gates and has been extended to other finite groups. Assumptions on having identical noise levels per gate have been lessened ([[Standard Randomised Benchmarking]]) and [[Randomised Benchmarking with confidence]] introduced. RB schemes have been generalized to other measures of quality, such as relative average gate fidelities ([[Interleaved Randomised Benchmarking]]), the unitarity [[Purity Benchmarking]], measures for losses [[Robust characterization of loss rates]], leakage [[Robust characterization of leakage errors]], addressibility [[ Characterization of addressability by randomized benchmarking]] or even tomographic schemes that combine data from multiple RB experiment.

==Protocols==
* [[Characterization of addressability by randomized benchmarking]]
* [[Interleaved Randomised Benchmarking]]
* [[Purity Benchmarking]]
* [[Randomised Benchmarking with confidence]]
* [[Robust characterization of leakage errors]]
* [[Robust characterization of loss rates]]
* [[Standard Randomised Benchmarking]]

==Properties==
* The figure of merits in different protocols are: average error rate, average fidelity of a noise quantum circuit, unitarity, measures for losses, leakage, addressibility and cross-talk.

==Related Papers==
* E.Knill et al (2007) arXiv:0707.0963: gate and time-independent noise model
* E. Mageson et al (2011) arXiv:1009.3639: multi-parameter model
* Magesan et al. PRL (2012): Interleaved Randomized Benchmarking
* Harper et al (2016) arXiv:1608.02943v2: Interleaved Randomised Benchmarking to estimate fidelity of T gates
* Wallman, Granade, Harper, F., NJP 2015: Purity benchmarking

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Randomised Benchmarking

2020-05-31T08:02:00Z

Rhea: /* Functionality Description */

==Functionality Description==
Randomized benchmarking refers to a collection of methods that aim to reliably estimating the magnitude of an average error of a quantum gate set in a robust fashion against state preparation and measurement error. It achieves this goal by applying sequences of feasible quantum gates of varying length, so that small errors are amplified with the sequence length leading

From a pragmatic point of view, RB protocols thereby define benchmarks that can be used to compare different digital quantum devices.In important instances ([[Standard Randomised Benchmarking]]), the benchmark can be related to the average gate fidelity, rendering RB protocols flexible certification tools. To this end, a group structure of the gate set is made use to achieve two goals: On the one hand, this is to control the theoretical prediction of error-free sequences.

RB is most prominently considered for Clifford gates and has been extended to other finite groups. Assumptions on having identical noise levels per gate have been lessened ([[Standard Randomised Benchmarking]]) and [[Randomised Benchmarking with confidence]] introduced. RB schemes have been generalized to other measures of quality, such as relative average gate fidelities ([[Interleaved Randomised Benchmarking]]), the unitarity [[Purity Benchmarking]], measures for losses [[Robust characterization of loss rates]], leakage [[Robust characterization of leakage errors]], addressibility [[ Characterization of addressability by randomized benchmarking]] or even tomographic schemes that combine data from multiple RB experiment.

==Protocols==
* [[Interleaved Randomised Benchmarking]]
* [[Purity Benchmarking]]
* [[Standard Randomised Benchmarking]]

==Properties==
* The figure of merits in different protocols are: average error rate, average fidelity of a noise quantum circuit, unitarity, measures for losses, leakage, addressibility and cross-talk.

==Related Papers==
* E.Knill et al (2007) arXiv:0707.0963: gate and time-independent noise model
* E. Mageson et al (2011) arXiv:1009.3639: multi-parameter model
* Magesan et al. PRL (2012): Interleaved Randomized Benchmarking
* Harper et al (2016) arXiv:1608.02943v2: Interleaved Randomised Benchmarking to estimate fidelity of T gates
* Wallman, Granade, Harper, F., NJP 2015: Purity benchmarking

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Randomised Benchmarking

2020-05-31T08:01:14Z

Rhea: /* Properties */

==Functionality Description==
Randomized benchmarking refers to a collection of methods that aim to reliably estimating the magnitude of an average error of a quantum gate set in a robust fashion against state preparation and measurement error. It achieves this goal by applying sequences of feasible quantum gates of varying length, so that small errors are amplified with the sequence length leading

From a pragmatic point of view, RB protocols thereby define benchmarks that can be used to compare different digital quantum devices.In important instances ([[Standard Randomised Benchmarking]]), the benchmark can be related to the average gate fidelity, rendering RB protocols flexible certification tools. To this end, a group structure of the gate set is made use to achieve two goals: On the one hand, this is to control the theoretical prediction of error-free sequences.

RB is most prominently considered for Clifford gates and has been extended to other finite groups. Assumptions on having identical noise levels per gate have been lessened ([[Standard Randomised Benchmarking]]) and [[Randomised Benchmarking with confidence]] introduced. RB schemes have been generalized to other measures of quality, such as relative average gate fidelities ([[Interleaved Randomised Benchmarking]]), the unitarity [[Purity Benchmarking]], measures for losses, leakage, addressibility and cross-talk or even tomographic schemes that combine data from multiple RB experiment.

==Protocols==
* [[Interleaved Randomised Benchmarking]]
* [[Purity Benchmarking]]
* [[Standard Randomised Benchmarking]]

==Properties==
* The figure of merits in different protocols are: average error rate, average fidelity of a noise quantum circuit, unitarity, measures for losses, leakage, addressibility and cross-talk.

==Related Papers==
* E.Knill et al (2007) arXiv:0707.0963: gate and time-independent noise model
* E. Mageson et al (2011) arXiv:1009.3639: multi-parameter model
* Magesan et al. PRL (2012): Interleaved Randomized Benchmarking
* Harper et al (2016) arXiv:1608.02943v2: Interleaved Randomised Benchmarking to estimate fidelity of T gates
* Wallman, Granade, Harper, F., NJP 2015: Purity benchmarking

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Certification Library

2020-05-31T07:57:50Z

Rhea:

{| class="wikitable"
!width="40%"|Technique
!width="100%"|Protocols
|-
|rowspan="2"|[[Building Trust]]||[[Self verification]]
|-
|[[Cross-Platform verification of Intermediate Scale Quantum Devices]]
|-
|rowspan="1"|[[Cross Entropy Benchmarking]]||[[Cross Entropy Benchmarking]]
|-
|rowspan="1"|[[Cycle Benchmarking]]||[[Cycle Benchmarking]]
|-
|rowspan="1"|[[Hamiltonian and Phase Estimation]]||[[Hamiltonian and Phase Estimation]]
|-
|rowspan="2"|[[Fidelity Estimation]]||[[Direct Fidelity Estimation]]
|-
|[[Estimate average Fidelity of quantum gates]]
|-
|rowspan="4"|[[Fidelity Witnessing]]||[[Fidelity witnesses for fermionic quantum simulations]]
|-
|[[Fidelity witnesses for multi-partite entangled states]]
|-
|[[Fidelity witnesses for Gaussian bosonic]]
|-
|[[Fidelity witnesses for Ground States of local Hamiltonians]]
|-
|rowspan="2"|[[Learning Techniques]]||[[Probably Approximately Correctly Learning (PAC)]]
|-
|[[Robust online Hamiltonian learning]]
|-
|rowspan="3"|[[Process Tomography]]||[[Compressed Quantum Process Tomography]]
|-
|[[Full Quantum Process Tomography with Linear inversion]]
|-
|[[Quantum Gate Set Tomography]]
|-
|rowspan="7"|[[Randomised Benchmarking]]||[[Characterization of addressability by randomized benchmarking]]
|-
|[[Interleaved Randomised Benchmarking]]
|-
|[[Purity Benchmarking]]
|-
|[[Randomised Benchmarking with confidence]]
|-
|[[Robust characterization of leakage errors]]
|-
|[[Robust characterization of loss rates]]
|-
|[[Standard Randomised Benchmarking]]
|-
|rowspan="8"|[[State Tomography]]||[[Compressed Sensing Tomography]]
|-
|[[Full Quantum state tomography with Linear Inversion]]
|-
|[[Full Quantum state tomography with Maximum Likelihood Estimation]]
|-
|[[Full Quantum state tomography with Bayesian mean estimation (BME)]]
|-
|[[Full Quantum state tomography using confidence regions]]
|-
|[[Many-body quantum state tomography with neural networks]]
|-
|[[Matrix Product State tomography]]
|-
|[[Tensor Network Tomography]]
|-
|rowspan="1"|[[Quantum Volume Technique]]||[[Quantum Volume Estimation]]
|-

Randomised Benchmarking

2020-05-31T07:52:17Z

Rhea: /* Protocols */

==Functionality Description==
Randomized benchmarking refers to a collection of methods that aim to reliably estimating the magnitude of an average error of a quantum gate set in a robust fashion against state preparation and measurement error. It achieves this goal by applying sequences of feasible quantum gates of varying length, so that small errors are amplified with the sequence length leading

From a pragmatic point of view, RB protocols thereby define benchmarks that can be used to compare different digital quantum devices.In important instances ([[Standard Randomised Benchmarking]]), the benchmark can be related to the average gate fidelity, rendering RB protocols flexible certification tools. To this end, a group structure of the gate set is made use to achieve two goals: On the one hand, this is to control the theoretical prediction of error-free sequences.

RB is most prominently considered for Clifford gates and has been extended to other finite groups. Assumptions on having identical noise levels per gate have been lessened ([[Standard Randomised Benchmarking]]) and [[Randomised Benchmarking with confidence]] introduced. RB schemes have been generalized to other measures of quality, such as relative average gate fidelities ([[Interleaved Randomised Benchmarking]]), the unitarity [[Purity Benchmarking]], measures for losses, leakage, addressibility and cross-talk or even tomographic schemes that combine data from multiple RB experiment.

==Protocols==
* [[Interleaved Randomised Benchmarking]]
* [[Purity Benchmarking]]
* [[Standard Randomised Benchmarking]]

==Properties==
* The noise model is assumed to be IID.
* This method is insensitive to the SPAM errors
* The figure of merit is average error rate, average fidelity of a noise quantum circuit.

==Related Papers==
* E.Knill et al (2007) arXiv:0707.0963: gate and time-independent noise model
* E. Mageson et al (2011) arXiv:1009.3639: multi-parameter model
* Magesan et al. PRL (2012): Interleaved Randomized Benchmarking
* Harper et al (2016) arXiv:1608.02943v2: Interleaved Randomised Benchmarking to estimate fidelity of T gates
* Wallman, Granade, Harper, F., NJP 2015: Purity benchmarking

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Randomised Benchmarking

2020-05-31T07:51:48Z

Rhea: /* Functionality Description */

==Functionality Description==
Randomized benchmarking refers to a collection of methods that aim to reliably estimating the magnitude of an average error of a quantum gate set in a robust fashion against state preparation and measurement error. It achieves this goal by applying sequences of feasible quantum gates of varying length, so that small errors are amplified with the sequence length leading

From a pragmatic point of view, RB protocols thereby define benchmarks that can be used to compare different digital quantum devices.In important instances ([[Standard Randomised Benchmarking]]), the benchmark can be related to the average gate fidelity, rendering RB protocols flexible certification tools. To this end, a group structure of the gate set is made use to achieve two goals: On the one hand, this is to control the theoretical prediction of error-free sequences.

RB is most prominently considered for Clifford gates and has been extended to other finite groups. Assumptions on having identical noise levels per gate have been lessened ([[Standard Randomised Benchmarking]]) and [[Randomised Benchmarking with confidence]] introduced. RB schemes have been generalized to other measures of quality, such as relative average gate fidelities ([[Interleaved Randomised Benchmarking]]), the unitarity [[Purity Benchmarking]], measures for losses, leakage, addressibility and cross-talk or even tomographic schemes that combine data from multiple RB experiment.

==Protocols==
* [[Standard Randomised Benchmarking]]
* [[Interleaved Randomised Benchmarking]]
* [[Purity Benchmarking]]

==Properties==
* The noise model is assumed to be IID.
* This method is insensitive to the SPAM errors
* The figure of merit is average error rate, average fidelity of a noise quantum circuit.

==Related Papers==
* E.Knill et al (2007) arXiv:0707.0963: gate and time-independent noise model
* E. Mageson et al (2011) arXiv:1009.3639: multi-parameter model
* Magesan et al. PRL (2012): Interleaved Randomized Benchmarking
* Harper et al (2016) arXiv:1608.02943v2: Interleaved Randomised Benchmarking to estimate fidelity of T gates
* Wallman, Granade, Harper, F., NJP 2015: Purity benchmarking

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Randomised Benchmarking

2020-05-30T19:23:45Z

Rhea: /* Properties */

==Functionality Description==
Randomized benchmarking refers to a collection of methods that aim to reliably estimating the magnitude of an average error of a quantum gate set in a robust fashion against state preparation and measurement error. One key [[figure of merit]] here is average gate fidelity. It achieves this goal by applying sequences of feasible quantum gates of varying length, so that small errors are amplified leading to reliable estimation.

==Protocols==
* [[Standard Randomised Benchmarking]]
* [[Interleaved Randomised Benchmarking]]
* [[Purity Benchmarking]]

==Properties==
* The noise model is assumed to be IID.
* This method is insensitive to the SPAM errors
* The figure of merit is average error rate, average fidelity of a noise quantum circuit.

==Related Papers==
* E.Knill et al (2007) arXiv:0707.0963: gate and time-independent noise model
* E. Mageson et al (2011) arXiv:1009.3639: multi-parameter model
* Magesan et al. PRL (2012): Interleaved Randomized Benchmarking
* Harper et al (2016) arXiv:1608.02943v2: Interleaved Randomised Benchmarking to estimate fidelity of T gates
* Wallman, Granade, Harper, F., NJP 2015: Purity benchmarking

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Cross-Platform verification of Intermediate Scale Quantum Devices

2020-05-30T19:18:57Z

Rhea:

This [https://arxiv.org/abs/1909.01282 protocol] is used to perform cross-platform verification of quantum simulators and quantum computers. This is used to directly measure the overlap and purities of two quantum states prepared in two different physical platforms and thus used to measure the fidelity of two possibly mixed states. This protocol infers the cross-platform fidelity of two quantum states from statistical correlations between the randomized measurements performed on the two different devices.

'''Tags:''' [[:Category: Certification protocol|Certification Protocol]], [[Cross-Platform Fidelity]], [[Building Trust]], [[Two devices]]

==Assumptions==
* There are no prior assumptions on the quantum states.
* The spin values for the two quantum devices are known.

==Outline==
The aim here to perform cross-platform verification by measuring the overlap of quantum states produced with two different experimental setups, potentially realized on very different physical platforms, without any prior assumptions on the quantum states themselves. This can be used to
whether two quantum devices have prepared the same quantum state. Here, the cross-platform fidelity is inferred from the statistical correlations between randomized measurements performed on the first and second device.

This protocol to measure the cross-platform fidelity of two quantum states requires only classical communication of random unitaries and measurement outcomes between the two platforms, with the experiments possibly taking place at very different points in time and space.

This protocol consists of the following steps:

* We start with two quantum devices which are based on different physical platforms, each consisting of two different spins. Two quantum operations are prepared in these quantum devices, which are each described by a density matrix.
* We find the reduced density matrices for the sub-systems of identical size for each device using partial trace operator over that sub-system.
* We apply a same random unitary is applied to the two quantum states. This random unitary is defined as the product of local random unitaries acting on all spins of the subsystem. Here, the local random unitaries are sampled independently from a [[unitary 2-design]] defined on the local Hilbert space and sent via classical communication to both devices.
* Now projective measurements in a computational basis are performed for both the systems.
* Repeating these measurements for the fixed random unitary provides us with the estimates of probability of measurement outcomes for the both the states.
* This entire procedure is then repeated for many different random unitaries.
* Finally we estimate the density matrix from the second order cross-correlations between the two platforms using the ensemble average of probabilities over random unitaries from the above procedure.
* The purities for the two sub systems are obtained as second-order auto-correlations of the probabilities.

==Notation==
* <math>N_M</math>: Finite number of projective measurements performed per random unitary
* <math>N_U</math>: Finite number of random unitaries used to infer overlap
* <math>\epsilon</math>: Fixed value of statistical error
* <math>S_1, S_2</math>: Two devices realised on different physical platforms
* <math>N_1, N_2</math>: Spins consisted in <math>S_1</math> and <math>S_2</math> respectively
* <math>U_1, U_2</math>: Quantum operation prepared in <math>S_1</math> and <math>S_2</math> respectively
* <math>\rho_1, \rho_2</math>: Density matrices of <math>U_1</math> and <math>U_2</math> respectively
* <math>\rho_{i,A_i}</math>: Reduced density matrices
* <math>A_i</math>: Sub system of identical size <math>N_{A}</math> where <math>A_i \subseteq S_i</math>
* <math>N_{A}</math>: Size of subset <math>A_i</math>. Subsets <math>A_1</math> and <math>A_2</math> have the equal size <math>N_{A}</math>
* <math>D_A</math>: Associated Hilbert space dimension, <math>D_A = d^{N_A}</math>
* <math>U_A</math>: Random unitary
* <math>U_k</math>: Local random unitaries acting on spins <math>k=1, .., N_A</math>. Here, the <math>U_k</math> are sampled independently from a [[unitary 2-design]] defined on the local Hilbert space <math>C^d</math>
* <math>k</math>: Spin
* <math>|s_A \rangle</math>: This denotes a string of possible measurement outcomes for spins. <math>|s_A \rangle =|s_1, ..., S_{N_A} \rangle </math>
* <math>P_U^{(i)}</math>: Estimate of probabilities of measurement outcome for different spins.
* <math>\overline{P_U^{(i)}(s_A)}</math>: The ensemble average over random unitaries of the form <math>U_A</math>
* <math>D[s_A, s_A']</math>: Hamming distance between two strings <math>s_A, s_A'</math> is defined as the number of spins where <math>s_k \neq s_k'</math> i.e. <math>D[s_A,s_A'] = |\{k \in \{ 1, .., N_A\}| s_k \neq s_k'\}|</math>
* <math>F_{max}(\rho_1, \rho_2)</math>: Cross platform fidelity between two quantum states

==Hardware Requirements==
* Two quantum devices on two different physical platforms
* Trusted Measurement device.
* Classical communication channel

==Properties==
* '''Figure of merit''': Cross-platform fidelity of two quantum states
* We can estimate the density matrix overlap of two quantum states here as well as their purities.
* The present protocol scales, although exponentially, much more favorably with the (sub)system size, allowing practical cross-platform verification for (sub)systems involving tens of qubits on state-of-the-art quantum devices.
* This protocol can be used to perform fidelity estimation towards known target theoretical states, as an experiment-theory comparison.
* In practice, from a finite number of projective measurements performed per random unitary (<math>N_M</math>), a statistical error of the estimated fidelity arises. With that, a finite number (<math>N_U</math>) of random unitaries used to infer overlap and purities can also cause a statistical error while estimating fidelity. Therefore, the scaling of the total number of experimental runs <math>N_MN_U</math>, which are required to reduce this statistical error below a fixed value of <math>\epsilon</math>
* In the regime <math>N_M \leq D_A</math> and <math>N_U \gg 1</math>, the average statistical error <math>|[F_{max}(\rho_A, \rho_A)]_e - 1| ~ 1/(N_M \sqrt{N_U})</math>. For unit target fidelity, the optimal allocation of the total measurement budget <math>N_U N_M</math> is thus to keep <math>N_U</math> small and fixed.
* The fidelity estimation of PR (entangled) states is thus less prone to statistical errors which we attribute to the fact that fluctuations across random unitaries are reduced due to the mixedness of the subsystems.
* The optimal allocation of <math>N_U</math> vs. <math>N_M</math> for given <math>N_UN_M</math> depends on the quantum states, in particular their fidelity and the allowed statistical error <math>\epsilon</math>,and is thus a priori not known
* In larger quantum systems, it gives access to the fidelities of all possible subsystems up to a given size – determined by the accepted statistical error and the measurement budget – and thus enables a fine-grained comparison of large quantum systems.

==Protocol Description==
'''Input''': <math>S_1, S_2</math> with spins <math>N_1, N_2</math>

'''Output''': Cross platform fidelity <math>F_{max}</math>

* Prepare <math>U_1, U_2</math> with density matrices <math>\rho_1, \rho_2</math> in <math>S_1</math> and <math>S_2</math> respectively.
* Denote <math>\rho_{i,A_i}</math> in <math>D_A</math> as <math>tr_{S_i / A_i} (\rho_i)</math> for <math>A_i \subseteq S_i (i = 1,2)</math>
* For <math>1, ...., N_u</math>
** For spin <math>k= 1, ..., N_A</math>:
*** Sample <math>U_k</math> independently from a [[unitary 2-design]]
*** Send <math>U_k</math> classically to <math>S_1</math> and <math>S_2</math>
** For <math>i= 1,2</math>:
*** Apply <math>U_A = \bigotimes_{k=1}^{N_A}U_k</math> to <math>\rho_{i,A_i}</math> in <math>S_i</math>
*** For <math>1, ..., N_M</math>:
**** Perform projective measurements in a computational basis in <math>S_i</math>
**** Obtain <math>|s_A \rangle</math>
*** Get estimates of probabilities <math>P_U^{(i)}(s_A) = Tr_{A_i}[U_A \rho_{i,A_i}U^{\dagger}_A |s_A\rangle\langle s_A|]</math>
* Obtain <math>\overline{P_U^{(i)}(s_A)}</math> and <math>\overline{P_U^{(j)}(s_A)}</math>
* Define Tr<math>[\rho_{i, A_i}, \rho_{j, A_j}] = d^{N_A}\sum_{s_A,s_A'} (-d)^{-D[s_A, s_A']}\overline{P_U^{(i)}(s_A)P_U^{(j)}(s_A')}</math>
* For density matrix overlap:
** Substitute <math>i = 1, j = 2</math>, to get Tr<math>[\rho_{1, A_1}, \rho_{2, A_2}]</math>
* For the purities of the quantum states:
** Substitute <math>i =j = 1</math> for Tr<math>\rho_{1, A_1}^2</math>
** Substitute <math>i =j = 2</math> for Tr<math>\rho_{2, A_2}^2</math>
* Calculate <math>F_{max}</math> using: <math>F_{max} = \frac{Tr[\rho_{1, A_1}, \rho_{2, A_2}]}{max\{Tr\rho_{1, A_1}^2, Tr\rho_{2, A_2}^2\}}</math>

==Further Information==
* In principle, the cross-platform fidelity can be determined from full quantum state tomography of the two quantum devices. However due to the exponential scaling with the (sub)system size, this approach is limited to only a few degrees of freedom, In contrast, as demonstrated below, the present protocol scales, although exponentially, much more favorably with the (sub)system size, allowing practical cross-platform verification for (sub)systems involving tens of qubits on state-of-the-art quantum devices.
* Significantly fewer measurements are required here than full quantum state tomography.

==Related Papers==
* A.Elben et al (2020) arXiv:1909.01282: Cross-Platform Verification of Intermediate Scale Quantum Devices

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Cross-Platform verification of Intermediate Scale Quantum Devices

2020-05-30T17:38:52Z

Rhea:

This [https://arxiv.org/abs/1909.01282 protocol] is used to perform cross-platform verification of quantum simulators and quantum computers. This is used to directly measure the overlap and purities of two quantum states prepared in two different physical platforms and thus used to measure the fidelity of two possibly mixed states. This protocol infers the cross-platform fidelity of two quantum states from statistical correlations between the randomized measurements performed on the two different devices.

'''Tags:''' [[:Category: Certification protocol|Certification Protocol]], [[Cross-Platform Fidelity]], [[Building Trust]], [[Two devices]]

==Assumptions==
* There are no prior assumptions on the quantum states.
* The spin values for the two quantum devices are known.

==Outline==
The aim here to perform cross-platform verification by measuring the overlap of quantum states produced with two different experimental setups, potentially realized on very different physical platforms, without any prior assumptions on the quantum states themselves. This can be used to
whether two quantum devices have prepared the same quantum state. Here, the cross-platform fidelity is inferred from the statistical correlations between randomized measurements performed on the first and second device.

This protocol to measure the cross-platform fidelity of two quantum states requires only classical communication of random unitaries and measurement outcomes between the two platforms, with the experiments possibly taking place at very different points in time and space.

This protocol consists of the following steps:

* We start with two quantum devices which are based on different physical platforms, each consisting of two different spins. Two quantum operations are prepared in these quantum devices, which are each described by a density matrix.
* We find the reduced density matrices for the sub-systems of identical size for each device using partial trace operator over that sub-system.
* We apply a same random unitary is applied to the two quantum states. This random unitary is defined as the product of local random unitaries acting on all spins of the subsystem. Here, the local random unitaries are sampled independently from a [[unitary 2-design]] defined on the local Hilbert space and sent via classical communication to both devices.
* Now projective measurements in a computational basis are performed for both the systems.
* Repeating these measurements for the fixed random unitary provides us with the estimates of probability of measurement outcomes for the both the states.
* This entire procedure is then repeated for many different random unitaries.
* Finally we estimate the density matrix from the second order cross-correlations between the two platforms using the ensemble average of probabilities over random unitaries from the above procedure.
* The purities for the two sub systems are obtained as second-order auto-correlations of the probabilities.

==Notation==
* <math>N_M</math>: Finite number of projective measurements performed per random unitary
* <math>N_U</math>: Finite number of random unitaries used to infer overlap
* <math>\epsilon</math>: Fixed value of statistical error
* <math>S_1, S_2</math>: Two devices realised on different physical platforms
* <math>N_1, N_2</math>: Spins consisted in <math>S_1</math> and <math>S_2</math> respectively
* <math>U_1, U_2</math>: Quantum operation prepared in <math>S_1</math> and <math>S_2</math> respectively
* <math>\rho_1, \rho_2</math>: Density matrices of <math>U_1</math> and <math>U_2</math> respectively
* <math>\rho_{i,A_i}</math>: Reduced density matrices
* <math>A_i</math>: Sub system of identical size <math>N_{A}</math> where <math>A_i \subseteq S_i</math>
* <math>N_{A}</math>: Size of subset <math>A_i</math>. Subsets <math>A_1</math> and <math>A_2</math> have the equal size <math>N_{A}</math>
* <math>D_A</math>: Associated Hilbert space dimension, <math>D_A = d^{N_A}</math>
* <math>U_A</math>: Random unitary
* <math>U_k</math>: Local random unitaries acting on spins <math>k=1, .., N_A</math>. Here, the <math>U_k</math> are sampled independently from a [[unitary 2-design]] defined on the local Hilbert space <math>C^d</math>
* <math>k</math>: Spin
* <math>|s_A \rangle</math>: This denotes a string of possible measurement outcomes for spins. <math>|s_A \rangle =|s_1, ..., S_{N_A} \rangle </math>
* <math>P_U^{(i)}</math>: Estimate of probabilities of measurement outcome for different spins.
* <math>\overline{P_U^{(i)}(s_A)}</math>: The ensemble average over random unitaries of the form <math>U_A</math>
* <math>D[s_A, s_A']</math>: Hamming distance between two strings <math>s_A, s_A'</math> is defined as the number of spins where <math>s_k \neq s_k'</math> i.e. <math>D[s_A,s_A'] = |\{k \in \{ 1, .., N_A\}| s_k \neq s_k'\}|</math>
* <math>F_{max}(\rho_1, \rho_2)</math>: Cross platform fidelity between two quantum states

==Hardware Requirements==
* Two quantum devices on two different physical platforms
* Trusted Measurement device.
* Classical communication channel

==Properties==
* '''Figure of merit''': Cross-platform fidelity of two quantum states
* We can estimate the density matrix overlap of two quantum states here as well as their purities.
* The present protocol scales, although exponentially, much more favorably with the (sub)system size, allowing practical cross-platform verification for (sub)systems involving tens of qubits on state-of-the-art quantum devices.
* This protocol can be used to perform fidelity estimation towards known target theoretical states, as an experiment-theory comparison.
* In practice, from a finite number of projective measurements performed per random unitary (<math>N_M</math>), a statistical error of the estimated fidelity arises. With that, a finite number (<math>N_U</math>) of random unitaries used to infer overlap and purities can also cause a statistical error while estimating fidelity. Therefore, the scaling of the total number of experimental runs <math>N_MN_U</math>, which are required to reduce this statistical error below a fixed value of <math>\epsilon</math>

==Protocol Description==
'''Input''': <math>S_1, S_2</math> with spins <math>N_1, N_2</math>

'''Output''': Cross platform fidelity <math>F_{max}</math>

* Prepare <math>U_1, U_2</math> with density matrices <math>\rho_1, \rho_2</math> in <math>S_1</math> and <math>S_2</math> respectively.
* Denote <math>\rho_{i,A_i}</math> in <math>D_A</math> as <math>tr_{S_i / A_i} (\rho_i)</math> for <math>A_i \subseteq S_i (i = 1,2)</math>
* For <math>1, ...., N_u</math>
** For spin <math>k= 1, ..., N_A</math>:
*** Sample <math>U_k</math> independently from a [[unitary 2-design]]
*** Send <math>U_k</math> classically to <math>S_1</math> and <math>S_2</math>
** For <math>i= 1,2</math>:
*** Apply <math>U_A = \bigotimes_{k=1}^{N_A}U_k</math> to <math>\rho_{i,A_i}</math> in <math>S_i</math>
*** For <math>1, ..., N_M</math>:
**** Perform projective measurements in a computational basis in <math>S_i</math>
**** Obtain <math>|s_A \rangle</math>
*** Get estimates of probabilities <math>P_U^{(i)}(s_A) = Tr_{A_i}[U_A \rho_{i,A_i}U^{\dagger}_A |s_A\rangle\langle s_A|]</math>
* Obtain <math>\overline{P_U^{(i)}(s_A)}</math> and <math>\overline{P_U^{(j)}(s_A)}</math>
* Define Tr<math>[\rho_{i, A_i}, \rho_{j, A_j}] = d^{N_A}\sum_{s_A,s_A'} (-d)^{-D[s_A, s_A']}\overline{P_U^{(i)}(s_A)P_U^{(j)}(s_A')}</math>
* For density matrix overlap:
** Substitute <math>i = 1, j = 2</math>, to get Tr<math>[\rho_{1, A_1}, \rho_{2, A_2}]</math>
* For the purities of the quantum states:
** Substitute <math>i =j = 1</math> for Tr<math>\rho_{1, A_1}^2</math>
** Substitute <math>i =j = 2</math> for Tr<math>\rho_{2, A_2}^2</math>
* Calculate <math>F_{max}</math> using: <math>F_{max} = \frac{Tr[\rho_{1, A_1}, \rho_{2, A_2}]}{max\{Tr\rho_{1, A_1}^2, Tr\rho_{2, A_2}^2\}}</math>

==Further Information==
* In principle, the cross-platform fidelity can be determined from full quantum state tomography of the two quantum devices. However due to the exponential scaling with the (sub)system size, this approach is limited to only a few degrees of freedom, In contrast, as demonstrated below, the present protocol scales, although exponentially, much more favorably with the (sub)system size, allowing practical cross-platform verification for (sub)systems involving tens of qubits on state-of-the-art quantum devices.

==Related Papers==
* A.Elben et al (2020) arXiv:1909.01282: Cross-Platform Verification of Intermediate Scale Quantum Devices

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Quantum Gate Set Tomography

2020-05-30T13:51:15Z

Rhea:

Quantum gate set tomography is a robust and powerful method for self-consistently characterizing an entire set of quantum logic gates on a black-box quantum device. This method arose from the observation that [[Quantum Process tomography]] is inaccurate in the presence of state preparation and measurement errors. This procedure estimates the complete process matrices of Pauli transfer matrices for the gate set based on experimental data.

'''Tags:''' [[Certification protocol]], [[Tomography]], [[Quantum process density matrix reconstruction]], [[Maximum Likelihood estimation]], [[Linear Inversion]], [[Gate set]]

==Assumptions==

* The selected measurement basis should be tomographically complete.
* For the calculation of the likelihood function we assume that the noise on the coincidence measurements has a Gaussian probability distribution. We also assume that each of our measurements is taken for the same amount of time.
* The SPAM gates are composed of gates in the gate set and therefore the minimal gate set must include sufficient gates to create a complete set of states and measurements.
* In the gate set, the first gate is always selected to be the null gate.
* We assume no knowledge about the gate set.

==Outline==

[[Quantum state tomography]] and [[Quantum process tomography]] assume that the initial states and measurements are known. But if the [[state preparation and measurement]] (SPAM) gates are faulty then the estimates provided using these techniques are faulty as well. Quantum gate set tomography solves this problem by including the SPAM gates self-consistently in the gate set to be estimated.

The goal of this method is to completely characterize a gate set, which includes an unknown set of gates and an initial state and 2-outcome [[POVM]]. For self-consistency, the SPAM gates are treated on the same footing as the original gates. The SPAM gates are composed of gates in the gate set and therefore the minimal gate set must include sufficient gates to create a complete set of states and measurements. When these gates are applied to our unknown fixed state and measurement, a complete set of initial states and final state is produced. The experimental requirement for GST is the ability to measure the expectation values for the gate set.

Two techniques are shown here, which are Linear inversion (LGST) and maximum likelihood estimation (MLE). The linear inversion protocol is fairly easy to implement numerically and is useful for providing quick diagnostics without resorting to more computationally intensive estimation via constrained optimization. The LGST estimate is not generally physical, the information obtained is somewhat qualitative, and hence MLE is preferred. LGST is a useful starting point for MLE. Since the starting point must be physical, while LGST is not, we should use the closest physical gate set to the LGST estimate.

This method consists of the following steps:

* Initialize the qubit to a particular state. In most systems, the natural choice for this state is the ground state of the qubit.
* For a particular choice of SPAM gates and a gate from the Gate set, the corresponding gate sequence is applied to the qubit. As the SPAM gates are composed of the gates from the gate set, the sequence only contains the gates from the gate set.
* The 2-outcome POVM is measured. This POVM is required to be a positive semi-definite Hermitian operator. The natural choice for this POVM in most systems is <math>|0\rangle\langle 0|</math>. (Sometimes <math>|1\rangle\langle 1|</math> is used)
* The above 1-3 steps are repeated a large number of times (10000 to 100000). For every repetition, the measurement is success (i.e., the measured state is <math>|0\rangle\langle 0|</math>) is recorded as 1 and the measurement failure (i.e., the measured state is not <math>|0\rangle\langle 0|</math>) is recorded as 0.
* The measurement results in the step above are averaged over the number of times the measurement was repeated and the expectation value is calculated.
* The above 1-5 steps are repeated for every SPAM gates and every gate in the gate set. This gives us the probability of all the possible gate set and SPAM gate combinations.
* Repeat steps 1-5 to measure the expectation values of only the SPAM gates. This is used to form a matrix known as the Gram matrix. If the first gate of the gate set is a null gate, then this step is not needed as this data already exists due to the step performed above.
* To find the gate set estimate from the measurement data, there are two techniques:
** Linear inversion method (LGST):
*** We should check that the gram matrix is invertible and after certain calculations with the inverted gram matrix, the gate set can be estimated. The LGST estimate is not generally physical, the information obtained is somewhat qualitative.
*** The gate set estimated in this way is in a different gauge from the actual gate set. To compensate, we transform to a more useful gauge. Since we do not know the actual gate set, only the target one, the most useful gauge is the one that brings the estimated gate set as close as possible based, on some distance metric, to the target. The gauge transformation is found by solving an optimization problem and the resulting gauge matrix is then applied to the gate set found in the step above.
** Maximum likelihood estimation (MLE): LGST is a useful starting point for MLE. The goal in this method is to find the true probabilities corresponding to the measurements, subject to physical constraints. In this method, the best estimate is found by fitting to experimental data a theoretical model of the probability of obtaining that data.
*** This method begins by parameterizing the estimate, state and measurement matrices in terms of a vector of parameters. A number of constraints reduce the total number of independent parameters.
*** Putting everything together, the probability estimates are written in terms of the parameter vector and Depending on the parameterization choice each of the gates and states is either a linear or quadratic function of its parameters. Each of the gates and states is either a linear or quadratic function of its parameters. The estimator is, therefore, a homogeneous function of order 5 (linear parameterization) or 10 (quadratic parameterization).
*** MLE proceeds by finding the set of parameters that minimizes an objective function. The objective, or likelihood function, is the probability distribution we assume produced the data.
*** There are two kinds of parameterizations for gates that are commonly used: The Pauli Process Matrix representation and The Pauli transfer Matrix Representation.

==Hardware Requirements==

* Measurement device.

==Notation==

* <math>d</math>: Dimension of Hilbert space
* <math>G</math>: Gate set. <math>G = \{G_0, G_1, .. G_K\}</math>. Generally <math>G_0 = {}</math> null gate.
* <math>F</math>: SPAM gate set. <math>F = \{F_0, ... , F_N\}</math>. <math>F</math> is composed of gates in <math>G</math>.
* <math>|\rho\rangle</math>: Initial quantum state. Generally the natural choice for <math>\rho</math> is <math>|0\rangle\langle 0|</math>
* <math>E</math>: 2-outcome POVM.
* <math>n</math>: Number of times the measurement is repeated.
* <math>n_r</math>: Result of <math>r^{th}</math> measurement.
* <math>m_{ijk}</math>: measurement of the expectation value <math>p_{ijk}</math>
* <math>p_{ijk}</math>: Expectation value. <math>p_{ijk} = \langle E|F_iG_kF_j|\rho\rangle</math>
* <math>g_{ij}</math>: Gram matrix
* <math>B_{ij}</math>: Gauge matrix
* <math>\vec{t}</math>: Parameter vector
* <math>\hat{G}(\vec{t})</math>: Parameterised estimate matrix. This has <math>d^4</math> parameters.
* <math>|\hat{p}(\vec{t})\rangle</math>: Parameterised state matrix. This has <math>d^2</math> parameters.
* <math>\langle \hat{E}(\vec{t})|</math>: Parameterised measurement matrix. This has <math>d^2</math> parameters.
* <math>l(\hat{G})</math>: likelihood function.
* <math>\sigma_{ijk}</math>: The sampling variance in the measurement <math>m</math>. <math>\sigma^2 = p(1-p)/n</math>
* <math>\chi_G</math>: Process matrix for gate <math>G</math> is defined in terms of the gate’s action on an arbitrary state <math>\rho</math> to produce a new state <math>G(p)</math>. <math>G(p) = \sum_{i,j=1}^{d^2}(\chi_G)_{ij}P_i\rho P_j</math>. <math>\chi</math> is a Hermitian positive semidefinite matrix, written in terms of <math>\chi = T^{\dagger}T</math>
* <math>T</math>: a lower-diagonal complex matrix, <math>
\hat{T}(t) = \begin{bmatrix}
t_1 & 0 & ... & 0 \\
t_{2^n + 1} + it_{2^n+2} & t_2 & ... & 0 \\
... & ... & ... & 0 \\
t_{4^n -1} + it_{4^n} & t_{4^n -3} + it_{4^n - 2} & t_{4^n - 5} + it_{4^n - 4} & t_{2^n}
\end{bmatrix}
</math>
* <math>P_i</math>: Pauli operator <math>i</math> acting on the state
* <math>R_G</math>: Pauli tranfer matrix for a gate G is defined in terms of the gate’s action on Pauli matrices. <math>(R_G)_{ij} = Tr\{P_iG(P_j)\}. |G(\rho)\rangle = R_G|\rho\rangle</math>

==Properties==

* Figure of merit: Gate set
* Multiple copies of the quantum state are required in this method.
* <math>E_j</math> should be tomographically complete.
* In contrast to QPT, it is not possible in GST to characterize one gate at a time. Instead, GST estimates every gate in the gate set simultaneously.
* SPAM gates are included in the gate set for self-consistency.
* The limitation of linear-inversion GST (or LGST) is that it does not constrain the estimates to be physical. However, LGST provides a convenient method for diagnosing gate errors and also gives a good starting point for the constrained maximum likelihood estimation.
* In the gate set, the first gate is generally selected to be the null gate.
* The simulated errors are of three different types, representing coherent and incoherent gate errors as well as intrinsic SPAM errors.

==Procedure Description==

'''Output''': Gate set, <math>G</math>

* Initialize system to <math>|\rho\rangle</math>
* For <math>i = 1, 2, ..., N</math>:
** For <math>j = 1, 2, ..., N</math>:
*** For <math>k = 1, 2, ..., K</math>:
**** For <math>r = 1, 2, ..., n</math>
***** Apply gate sequence <math>F_i \circ G_k \circ F_j</math>
***** Measure with POVM <math>E</math>, get <math>n_r = 1</math> or <math>n_r = 0</math>
**** <math>m_{ijk} = \sum_{r=1}^{n} \frac{n_r}{n}</math>
**** <math>(\tilde{G}_k)_{ij} = p_{ijk} = m_{ijk} = \langle E|F_iG_kF_j|\rho\rangle</math>
**** if <math>k==0</math> (null gate), <math>g_{ij} = \langle E|F_iF_j|\rho\rangle</math>
** <math>|\tilde{\rho}\rangle_i = \langle E|F_i|\rho\rangle</math>
** <math>B_{ij} = \langle i |F_j |\rho\rangle</math>
* For Linear inversion:
** Check that the Gram matrix <math>g_{ij}</math> is non singular, so that it may be inverted
** For the gate set, the estimate is <math>|\hat{\rho}\rangle = g^{-1}|\tilde{\rho}\rangle, \langle \hat{E}| = \langle \tilde{E} |, \hat{G_k} = g^{-1} \tilde{G}_k</math>
** Apply gauge optimization, by minimizing:
<div style="text-align: center;"><math>\hat{B}^* = argmin_{\hat{B}}\sum^{K+1}_{k=1}Tr\{(\hat{G}_k - \hat{B}^{-1}T_k\hat{B})^T(\hat{G}_k - \hat{B}^{-1}T_k\hat{B})\}
</math></div>
** Final gate set is: <math>\hat{G}_k^{*} = \hat{B}^*\hat{G}_k(\hat{B}^*)^{-1}, |\hat{p}^*\rangle = \hat{B}^*|\hat{p}\rangle, \langle \hat{E}^*| = \langle \hat{E}|(\hat{B}^*)^{-1}</math>
* For Maximum Likelihood Estimation:
** <math>\hat{p}_{ijk} = \langle \hat{E}(\vec{t})|\hat{F}_i(\vec{t})\hat{G}_k(\vec{t})\hat{F}_j(\vec{t})|\hat{\rho}(\vec{t})\rangle</math>
** Find the set of parameters <math>\vec{t}</math> where <math>l(\hat{G})</math> is minimized.
<div style="text-align: center;">Minimize: <math>
l(\hat{G}) = \sum_{ijk} (m_{ijk} - \hat{p}_{ijk}(\vec{t}))^2 / \sigma^2_{ijk}
</math></div>
** Minimize the above function with two different commonly used types of parameterisation of gates:
*** Pauli Process Matrix Optimization problem:
<div style="text-align: center;">Minimize: <math>
l(\hat{G}) = \sum_{ijk} (m_{ijk} - \sum_{mnrstu}(\chi_{F_i})_{tu} (\chi_{G_k})_{rs} (\chi_{F_j})_{mn} Tr\{EP_t P_r P_m \rho P_n P_s P_u\} )^2 / \sigma^2_{ijk}
</math></div>
<div style="text-align: center;">Subject to: <math>
\sum_{mn}(\chi_G)_{mn}Tr\{P_mP_rP_n\} - \delta_{0r} = 0, (r = 1, .. , d^2), \forall G \in Gateset
</math></div>
<div style="text-align: center;"><math>
Tr\{\rho\} = 1
</math></div>
<div style="text-align: center;"><math>
1 - E \geqslant 0
</math></div>
*** Pauli Transfer Matrix Representation
<div style="text-align: center;"> Minimize: <math>
l(\hat{G}) = \sum_{ijk} (m_{ijk} - \langle \hat{E}(\vec{t})|\hat{R}_{F_i}(\vec{t})\hat{R}_{G_k}(\vec{t})\hat{R}_{F_j}(\vec{t})|\hat{\rho}(\vec{t})\rangle )^2 / \sigma^2_{ijk}
</math></div>
<div style="text-align: center;">Subject to: <math>
\rho_G = \frac{1}{d^2} \sum_{i,j=1}^{d^2}(R_G)_{ij}P^T_G \otimes P_i \geqslant 0, \forall G \in Gateset
</math></div>
<div style="text-align: center;"><math>
(R_G)_{0i} = \delta_{0i}, \forall G \in Gateset
</math></div>
<div style="text-align: center;"><math>
(R_G)_{ij} \in [-1, 1], \forall G, i, j
</math></div>
<div style="text-align: center;"><math>
Tr\{\rho\} = 1
</math></div>
<div style="text-align: center;"><math>
1 - E \geqslant 0
</math></div>

==Further Information==

==Related Papers==
* Daniel Greenbaum (2015) arXiv:1509.02921v1: Introduction to Quantum Gate Set Tomography

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Cross-Platform verification of Intermediate Scale Quantum Devices

2020-05-30T11:57:25Z

Rhea: Created page with "This [https://arxiv.org/abs/1909.01282 protocol] is used to perform cross-platform verification of quantum simulators and quantum computers. This is used to directly measure t..."

Certification Library

2020-05-29T17:46:24Z

Rhea:

{| class="wikitable"
!width="40%"|Technique
!width="100%"|Protocols
|-
|rowspan="2"|[[Building Trust]]||[[Self verification]]
|-
|[[Cross-Platform verification of Intermediate Scale Quantum Devices]]
|-
|rowspan="1"|[[Cross Entropy Benchmarking]]||[[Cross Entropy Benchmarking]]
|-
|rowspan="1"|[[Cycle Benchmarking]]||[[Cycle Benchmarking]]
|-
|rowspan="1"|[[Hamiltonian and Phase Estimation]]||[[Hamiltonian and Phase Estimation]]
|-
|rowspan="2"|[[Fidelity Estimation]]||[[Direct Fidelity Estimation]]
|-
|[[Estimate average Fidelity of quantum gates]]
|-
|rowspan="4"|[[Fidelity Witnessing]]||[[Fidelity witnesses for fermionic quantum simulations]]
|-
|[[Fidelity witnesses for multi-partite entangled states]]
|-
|[[Fidelity witnesses for Gaussian bosonic]]
|-
|[[Fidelity witnesses for Ground States of local Hamiltonians]]
|-
|rowspan="2"|[[Learning Techniques]]||[[Probably Approximately Correctly Learning (PAC)]]
|-
|[[Robust online Hamiltonian learning]]
|-
|rowspan="3"|[[Process Tomography]]||[[Compressed Quantum Process Tomography]]
|-
|[[Full Quantum Process Tomography with Linear inversion]]
|-
|[[Quantum Gate Set Tomography]]
|-
|rowspan="3"|[[Randomised Benchmarking]]||[[Interleaved Randomised Benchmarking]]
|-
|[[Purity Benchmarking]]
|-
|[[Standard Randomised Benchmarking]]
|-
|rowspan="8"|[[State Tomography]]||[[Compressed Sensing Tomography]]
|-
|[[Full Quantum state tomography with Linear Inversion]]
|-
|[[Full Quantum state tomography with Maximum Likelihood Estimation]]
|-
|[[Full Quantum state tomography with Bayesian mean estimation (BME)]]
|-
|[[Full Quantum state tomography using confidence regions]]
|-
|[[Many-body quantum state tomography with neural networks]]
|-
|[[Matrix Product State tomography]]
|-
|[[Tensor Network Tomography]]
|-
|rowspan="1"|[[Quantum Volume Technique]]||[[Quantum Volume Estimation]]
|-

Certification Library

2020-05-29T17:22:25Z

Rhea:

{| class="wikitable"
!width="40%"|Technique
!width="100%"|Protocols
|-
|rowspan="2"|[[Building Trust]]||[[Self verification]]
|-
|[[Cross-Platform verification]]
|-
|rowspan="1"|[[Cross Entropy Benchmarking]]||[[Cross Entropy Benchmarking]]
|-
|rowspan="1"|[[Cycle Benchmarking]]||[[Cycle Benchmarking]]
|-
|rowspan="1"|[[Hamiltonian and Phase Estimation]]||[[Hamiltonian and Phase Estimation]]
|-
|rowspan="2"|[[Fidelity Estimation]]||[[Direct Fidelity Estimation]]
|-
|[[Estimate average Fidelity of quantum gates]]
|-
|rowspan="4"|[[Fidelity Witnessing]]||[[Fidelity witnesses for fermionic quantum simulations]]
|-
|[[Fidelity witnesses for multi-partite entangled states]]
|-
|[[Fidelity witnesses for Gaussian bosonic]]
|-
|[[Fidelity witnesses for Ground States of local Hamiltonians]]
|-
|rowspan="2"|[[Learning Techniques]]||[[Probably Approximately Correctly Learning (PAC)]]
|-
|[[Robust online Hamiltonian learning]]
|-
|rowspan="3"|[[Process Tomography]]||[[Compressed Quantum Process Tomography]]
|-
|[[Full Quantum Process Tomography with Linear inversion]]
|-
|[[Quantum Gate Set Tomography]]
|-
|rowspan="3"|[[Randomised Benchmarking]]||[[Interleaved Randomised Benchmarking]]
|-
|[[Purity Benchmarking]]
|-
|[[Standard Randomised Benchmarking]]
|-
|rowspan="8"|[[State Tomography]]||[[Compressed Sensing Tomography]]
|-
|[[Full Quantum state tomography with Linear Inversion]]
|-
|[[Full Quantum state tomography with Maximum Likelihood Estimation]]
|-
|[[Full Quantum state tomography with Bayesian mean estimation (BME)]]
|-
|[[Full Quantum state tomography using confidence regions]]
|-
|[[Many-body quantum state tomography with neural networks]]
|-
|[[Matrix Product State tomography]]
|-
|[[Tensor Network Tomography]]
|-
|rowspan="1"|[[Quantum Volume Technique]]||[[Quantum Volume Estimation]]
|-

Certification Library

2020-05-29T16:55:44Z

Rhea:

{| class="wikitable"
!width="40%"|Technique
!width="100%"|Protocols
|-
|rowspan="1"|[[Cross Entropy Benchmarking]]||[[Cross Entropy Benchmarking]]
|-
|rowspan="1"|[[Cycle Benchmarking]]||[[Cycle Benchmarking]]
|-
|rowspan="1"|[[Hamiltonian and Phase Estimation]]||[[Hamiltonian and Phase Estimation]]
|-
|rowspan="2"|[[Fidelity Estimation]]||[[Direct Fidelity Estimation]]
|-
|[[Estimate average Fidelity of quantum gates]]
|-
|rowspan="4"|[[Fidelity Witnessing]]||[[Fidelity witnesses for fermionic quantum simulations]]
|-
|[[Fidelity witnesses for multi-partite entangled states]]
|-
|[[Fidelity witnesses for Gaussian bosonic]]
|-
|[[Fidelity witnesses for Ground States of local Hamiltonians]]
|-
|rowspan="2"|[[Learning Techniques]]||[[Probably Approximately Correctly Learning (PAC)]]
|-
|[[Robust online Hamiltonian learning]]
|-
|rowspan="3"|[[Process Tomography]]||[[Compressed Quantum Process Tomography]]
|-
|[[Full Quantum Process Tomography with Linear inversion]]
|-
|[[Quantum Gate Set Tomography]]
|-
|rowspan="3"|[[Randomised Benchmarking]]||[[Interleaved Randomised Benchmarking]]
|-
|[[Purity Benchmarking]]
|-
|[[Standard Randomised Benchmarking]]
|-
|rowspan="8"|[[State Tomography]]||[[Compressed Sensing Tomography]]
|-
|[[Full Quantum state tomography with Linear Inversion]]
|-
|[[Full Quantum state tomography with Maximum Likelihood Estimation]]
|-
|[[Full Quantum state tomography with Bayesian mean estimation (BME)]]
|-
|[[Full Quantum state tomography using confidence regions]]
|-
|[[Many-body quantum state tomography with neural networks]]
|-
|[[Matrix Product State tomography]]
|-
|[[Tensor Network Tomography]]
|-
|rowspan="1"|[[Quantum Volume Technique]]||[[Quantum Volume Estimation]]
|-

Certification Library

2020-05-29T16:15:27Z

Rhea:

{| class="wikitable"
!width="40%"|Technique
!width="100%"|Protocols
|-
|rowspan="1"|[[Cross Entropy Benchmarking]]||[[Cross Entropy Benchmarking]]
|-
|rowspan="1"|[[Cycle Benchmarking]]||[[Cycle Benchmarking]]
|-
|rowspan="1"|[[Hamiltonian and Phase Estimation]]||[[Hamiltonian and Phase Estimation]]
|-
|rowspan="1"|[[Fidelity Estimation]]||[[Direct Fidelity Estimation]]
|-
|rowspan="1"|[[Fidelity Witnessing]]||[[Fidelity witnesses for fermionic quantum simulations]]
|-
|rowspan="3"|[[Process Tomography]]||[[Compressed Quantum Process Tomography]]
|-
|[[Full Quantum Process Tomography with Linear inversion]]
|-
|[[Quantum Gate Set Tomography]]
|-
|rowspan="3"|[[Randomised Benchmarking]]||[[Interleaved Randomised Benchmarking]]
|-
|[[Purity Benchmarking]]
|-
|[[Standard Randomised Benchmarking]]
|-
|rowspan="7"|[[State Tomography]]||[[Compressed Sensing Tomography]]
|-
|[[Full Quantum state tomography with Linear Inversion]]
|-
|[[Full Quantum state tomography with Maximum Likelihood Estimation]]
|-
|[[Full Quantum state tomography with Bayesian mean estimation (BME)]]
|-
|[[Full Quantum state tomography using confidence regions]]
|-
|[[Matrix Product State tomography]]
|-
|[[Tensor Network Tomography]]
|-
|rowspan="1"|[[Quantum Volume Technique]]||[[Quantum Volume Estimation]]
|-

Certification Library

2020-05-29T16:12:49Z

Rhea:

{| class="wikitable"
!width="40%"|Technique
!width="100%"|Protocols
|-
|rowspan="1"|[[Cross Entropy Benchmarking]]||[[Cross Entropy Benchmarking]]
|-
|rowspan="1"|[[Cycle Benchmarking]]||[[Cycle Benchmarking]]
|-
|rowspan="1"|[[Hamiltonian and Phase Estimation]]||[[Hamiltonian and Phase Estimation]]
|-
|rowspan="1"|[[Fidelity Estimation]]||[[Direct Fidelity Estimation]]
|-
|rowspan="1"|[[Fidelity Witnessing]]||[[Fidelity witnesses for fermionic quantum simulations]]
|-
|rowspan="2"|[[Process Tomography]]||[[Compressed Quantum Process Tomography]]
|-
|[[Full Quantum Process Tomography with Linear inversion]]
|-
|[[Quantum Gate Set Tomography]]
|-
|rowspan="3"|[[Randomised Benchmarking]]||[[Interleaved Randomised Benchmarking]]
|-
|[[Purity Benchmarking]]
|-
|[[Standard Randomised Benchmarking]]
|-
|rowspan="7"|[[State Tomography]]||[[Compressed Sensing Tomography]]
|-
|[[Full Quantum state tomography with Linear Inversion]]
|-
|[[Full Quantum state tomography with Maximum Likelihood Estimation]]
|-
|[[Full Quantum state tomography with Bayesian mean estimation (BME)]]
|-
|[[Full Quantum state tomography using confidence regions]]
|-
|[[Matrix Product State tomography]]
|-
|[[Tensor Network Tomography]]
|-
|rowspan="1"|[[Quantum Volume Technique]]||[[Quantum Volume Estimation]]
|-

Interleaved Randomised Benchmarking

2020-03-28T12:52:41Z

Rhea: /* Properties */

[https://arxiv.org/abs/1203.4550 Interleaved Randomized benchmarking] is a scalable experimental protocol for estimating the average error of individual quantum computational gates. This protocol consists of interleaving random Clifford gates between the gate of interest and provides an estimate as well as theoretical bounds for the average error of the gate under test, so long as the average noise variation over all Clifford gates is small. This technique takes into account both state preparation and measurement errors and is scalable in the number of qubits.

'''Tags:''' [[:Category: Certification protocol|Certification Protocol]], [[Randomised Benchmarking]], Clifford group

==Assumptions==
* The measurements performed are trusted.
* Noise model can be assumed to be gate and time-dependent or gate and time-independent.
* The noise model is independent and identically distributed (IID).

==Outline==
[[Standard Randomised Benchmarking]] method involves applying many random sequences of gates of varying lengths to a standard initial state. Each sequence ends with a randomized measurement that determines whether the correct final state was obtained. The average computationally relevant error per gate is obtained from the increase in error probability of the final measurements as a function of sequence length. The random gates are taken from the [[Clifford group]]. The restriction to the Clifford group ensures that the measurements can be of one-qubit Pauli operators that yield at least one deterministic one-bit answer in the absence of errors.

The multi-qubit RB protocol described in Standard Randomised Benchmarking is restricted to benchmark only the full [[Clifford group]] on <math>n</math> qubits. While this provides a significant step towards scalable benchmarking of a quantum information processor, it is desirable in many cases to benchmark individual gates in Clifford group rather than the entire set. Interleaving randomised benchmarking is a protocol which consists of interleaving random gates between the gate of interest, which is used to estimate the average error of individual quantum computational gates.

To benchmark a specific Clifford element (an individual gate), the following steps are involved:

'''Step 1''': Implement [[Standard Randomised Benchmarking]] to get a model for the fidelity and to calculate the average error rate

* A fixed sequence length is selected at random. A random sequence of this length is chosen from the Clifford group.
* The operations are applied to the initial state corresponding to the selected sequence and then a final operator is applied which inverts all the previous operations.
* The final state is then measured to check if it matches the initial state. This process is performed several times with the same sequence to estimate the survival probability (the probability that the final state which returns to its initial state).
* Other random sequences of the same fixed sequence length are picked and the above-mentioned process is repeated to calculate the corresponding survival probability. This is then used to calculate the average survival probability for the sequence length.
* The same procedure is repeated for multiple different randomly selected sequence lengths.
* The observed survival probabilities are then plotted against the sequence length and then this is fit to an exponential decay curve, which is used to estimate the depolarizing parameter and sequence fidelity and also to calculate the average error rate which is the metric for randomized benchmarking.

'''Step 2''': Procedure to estimate the new sequence fidelity by including the Clifford element to be benchmarked in the sequence

* Now, for a random fixed sequence length, choose a sequence where the first Clifford element is selected uniformly at random from the Clifford group and the second element is always chosen to be the specific Clifford element we want to benchmark.
* Final gate is chosen to be the inverse of the composition mentioned in the step above. The final state is then measured to check if it matches the initial state. This process is performed several times with the same sequence to estimate the survival probability (the probability that the final state which returns to its initial state).
* Other random sequences of the same fixed sequence length are picked and the above-mentioned process is repeated to calculate the corresponding survival probability. This is then used to calculate the average survival probability for the sequence length.
* The same procedure is repeated for multiple different randomly selected sequence lengths.
* The observed survival probabilities are then plotted against the sequence length, to obtain a zeroth or first-order model of the new sequence fidelity, from which the new depolarizing parameter is estimated.

'''Step 3''': Estimate the gate error of the selected Clifford element to be benchmarked

* From the values obtained for the depolarizing parameter and the new depolarizing parameter, using the average gate fidelity, a point estimate is obtained for the gate error. The gate error would lie in a certain range of this estimate. One interpretation of this error is that it arises from imperfect random gates.

==Hardware Requirements==
* Quantum computational resources to perform Clifford gates.
* Trusted Measurement device.

==Notation==
* <math>p</math>: Depolarizing parameter
* <math>p_{\bar{C}}</math>: New depolarizing parameter for the specific Clifford element to be benchmarked
* <math>d</math>: Dimension of Hilbert space
* <math>F_{avg}</math>: Average fidelity, <math>F_{avg} = p + \frac{1-p}{d}</math>
* <math>r</math>: Average error rate, <math>r = 1- F_{avg}, r = \frac{(d-1)(1-p)}{d}</math>
* <math>m</math>: Selected sequence length
* <math>K_m</math>: Total randomly selected sequence of <math>m</math> sequence length
* Clif<math>_n</math>: Clifford group
* C: Selected Clifford element to be benchmarked
* C<math>_i</math>: Random element of Clifford group
* <math>S_{(i_1, ...,i_m)}</math> = <math>S_{\mathbf{i_m}}</math>: Random sequence of operations of length <math>m</math>
* <math>\gamma</math>: Superoperator representing the sequence with alternating <math>C</math>
* <math>M</math>: Number of different data points to get the error model
* <math>\Lambda_{i,j}</math>: Implementation of C<math>_i</math> at time j (1 <math>\leq</math> j <math>\leq</math> M) results in this error map. <math>\Lambda_{i,1}, ..., \Lambda_{i,M}</math> are the different time-dependent noise operators affecting C<math>_i</math>.
* <math>\Lambda_{C}</math>: Associated noise operator of the Clifford element <math>C</math>
* <math>r^{est}_C</math>: The gate error of <math>\Lambda_{C}</math>
* <math>E</math>: Error range of <math>r^{est}_C</math>
* <math>|\psi\rangle</math>: initial state
* <math>E_{\psi}</math>: POVM element which takes into account the measurement error.
* <math>F_{seq}(m, \psi) = Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>: Survival probability of a sequence. <math>\rho_\psi</math> is a quantum state that takes into account errors in preparing <math>\langle \psi |\psi \rangle</math>
* <math>F_g^{(0)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time independent error model
* <math>F_g^{(1)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time dependent error model. In this model, the parameter <math>(q-p^2)</math> is a measure of the degree of gate-dependence in the error.
* <math>F_\bar{g}^{(0)}(m, |\psi\rangle)</math>: New zeroth order averaged sequence fidelity for <math>C</math>
* <math>F_\bar{g}^{(1)}(m, |\psi\rangle)</math>: New first order Averaged sequence fidelity for <math>C</math>
* <math>A_0, B_0</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time independent error model
* <math>A_1, B_1, C_1</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time dependent error model.
* <math>R_{m+1}</math>: <math>\frac{1}{|Clif_n|}\sum_i\Lambda_{i, m+1} \otimes (C_i \otimes \Lambda \otimes C_i^{\dagger})</math>

==Properties==
* '''Figure of merit''': average gate error
* This protocol is used to estimate the average error of individual quantum computational gate
* In the limits of either perfect random gates or that the average error of all gates is depolarizing, this protocol estimates the gate error perfectly
* In the completely general case where the random gates have arbitrary errors with small average variation, this protocol provides explicit bounds for the error of the gate. These bounds give direct information regarding the quality of computational gates and thus useful information about reaching thresholds for fault-tolerant quantum computation.
* This is a scalable protocol with the time complexity <math>O(n^4)</math>
* The errors which are considered here are State preparation and measurement errors, error on the final gate, which are gate and time-independent errors. Gate and time-dependent errors can also be taken into consideration. This method is insensitive to SPAM error.
* The random gates used to benchmark the specific Clifford gate are picked from the Clifford group.
* For noise estimation, the uniform probability distribution over Clifford group comprises a [[unitary 2-design]].

==Procedure Description==

'''Step 1''': Standard Randomised Benchmarking

'''Output''': Figure of merit: <math>r</math>

* For <math>1, 2, ..., M</math>:
** Pick random sequence length <math>m</math>
** For <math>k = 1, 2, ..., K_m</math> sequences:
*** For <math>j = 1, 2 ..., m+1</math>:
**** If <math>j == m+1</math>, apply inverse operator of previous operations
**** else, apply random operation C<math>_i</math>
*** Thus, <math>S_{\mathbf{i_m}} = \bigotimes^{m+1}_{j=1} (\Lambda_{(i_j, j)} \circ C_{i_j})</math> and <math>i_{m+1}</math> is uniquely determined by <math>(i_1, ...,i_m)</math>
*** Measure survival probability <math>Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>
** Estimate average survival probability <math>Tr[E_{\psi}S_{\mathbf{K_m}}(\rho_\psi)]</math> over all <math>K_m</math> sequences, where <math>S_{\mathbf{K_m}} = \frac{1}{K_m}\sum_{i_m} S_{i_m}</math>
* Fit the results for the averaged sequence fidelity for all <math>m</math> into the models:
** For gate and time independent error model:
*** <math>F_g^{(0)}(m, |\psi\rangle) = A_0p^m + B_0</math>
** For gate and time dependent error model:
*** <math>F_g^{(1)}(m, |\psi\rangle) = A_1p^m + B_1 + C_1(m-1)(q-p^2)p^{m-2}</math>
* <math>p</math> is extracted from the model and <math>r</math> is estimated, <math>r = \frac{(d-1)(1-p)}{d}</math>

'''Step 2''': Estimate gate error of selected Clifford element C

'''Input''': C

'''Output''': gate error of <math>\Lambda_{C}</math>: <math>r^{est}_C</math>

* For <math>1, 2, ..., M</math>:
** Pick random sequence length <math>m</math>
** For <math>k = 1, 2, ..., K_m</math> sequences:
*** For <math>j = 1, 2 ..., m+1</math>:
**** If <math>j == m+1</math>, apply inverse operator of previous operations
**** else If <math>j%2==1</math>, apply random operation C<math>_i</math>
**** else, apply C
*** Thus <math>\gamma = \Lambda_{i_{m+1}} +</math> C<math>_{i_{m+1}} (\bigotimes^{m+1}_{j=1}[C \circ \Lambda_C \circ \Lambda_{i_j} \circ C_{i_j}])</math>
*** Measure the survival probability <math>Tr[E_{\psi}\gamma_{\mathbf{i_m}}(\rho_\psi)]</math>
** Estimate average survival probability <math>Tr[E_{\psi}\gamma_{\mathbf{K_m}}(\rho_\psi)]</math> over all <math>K_m</math> sequences, where <math>\gamma_{\mathbf{K_m}} = \frac{1}{K_m}\sum_{i_m} \gamma_{i_m}</math>
* Fit the results for the averaged sequence fidelity for all <math>m</math> into the models, to find <math>p_{\bar{C}}</math>:
** For gate and time independent error model:
*** <math>F_\bar{g}^{(0)}(m, |\psi\rangle) = A_0p_{\bar{C}}^m + B_0</math>
** For gate and time dependent error model:
*** <math>F_\bar{g}^{(1)}(m, |\psi\rangle) = A_1p_{\bar{C}}^m + B_1 + C_1(m-1)(q-p_{\bar{C}}^2)p_{\bar{C}}^{m-2}</math>
* Estimate <math>r^{est}_C = \frac{(d-1)(1-p_{\bar{C}}/p)}{d}</math>
* <math>r^{est}_C</math> lies in the range <math>[r^{est}_C-E, r^{est}_C+E]</math>, where <math>E = min (\frac{(d-1)[|p-p_{\bar{C}}/p| + (1-p)]}{d}, \frac{2(d^2-1)(1-p)}{pd^2} + \frac{4\sqrt{1-p}\sqrt{d^2-1}}{p})</math>

==Further Information==
* Fitting models are described and derived as seen in [https://arxiv.org/abs/1109.6887v2 E. Mageson et al]. The coefficients derived are:
** <math>A_0</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
** <math>B_0</math> = Tr<math>[E_\psi \Lambda(\frac{\mathbb{1}}{d})]</math>
** <math>A_1</math> = Tr<math>[E_\psi \Lambda(\frac{Q_1\rho_\psi}{p} - \rho_\psi + \frac{(p-1)\mathbb{1}}{pd})]</math> + Tr<math>[E_\psi R_{m+1}(\frac{\rho_\psi}{p} - \frac{\mathbb{1}}{pd})]</math>
** <math>B_1</math> = Tr<math>[E_\psi R_{m+1}(\frac{\mathbb{1}}{d})]</math>
** <math>C_1</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
* The case where Randomized benchmarking fails: Suppose the noise is time dependent and for each <math>i, \Lambda_i = C_i^{\dagger}</math>. Then <math>F_g(m, \psi) = 1</math> for every <math>m</math> even though there is a substantial error on each <math>C_i</math> and so benchmarking fails.
* Wallman, Granade, Harper, F., NJP 2015 [[Purity benchmarking]]: A unitarity can be estimated via purity benchmarking, which is an RB-like experiment that estimates a decay rate.

==Related Papers==
* E.Knill et al (2007) arXiv:0707.0963: gate and time-independent noise model
* E. Mageson et al (2011) arXiv:1009.3639: multi-parameter model
* Magesan et al. PRL (2012): Interleaved Randomized Benchmarking

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Interleaved Randomised Benchmarking

2020-03-28T12:39:59Z

Rhea:

[https://arxiv.org/abs/1203.4550 Interleaved Randomized benchmarking] is a scalable experimental protocol for estimating the average error of individual quantum computational gates. This protocol consists of interleaving random Clifford gates between the gate of interest and provides an estimate as well as theoretical bounds for the average error of the gate under test, so long as the average noise variation over all Clifford gates is small. This technique takes into account both state preparation and measurement errors and is scalable in the number of qubits.

'''Tags:''' [[:Category: Certification protocol|Certification Protocol]], [[Randomised Benchmarking]], Clifford group

==Assumptions==
* The measurements performed are trusted.
* Noise model can be assumed to be gate and time-dependent or gate and time-independent.
* The noise model is independent and identically distributed (IID).

==Outline==
[[Standard Randomised Benchmarking]] method involves applying many random sequences of gates of varying lengths to a standard initial state. Each sequence ends with a randomized measurement that determines whether the correct final state was obtained. The average computationally relevant error per gate is obtained from the increase in error probability of the final measurements as a function of sequence length. The random gates are taken from the [[Clifford group]]. The restriction to the Clifford group ensures that the measurements can be of one-qubit Pauli operators that yield at least one deterministic one-bit answer in the absence of errors.

The multi-qubit RB protocol described in Standard Randomised Benchmarking is restricted to benchmark only the full [[Clifford group]] on <math>n</math> qubits. While this provides a significant step towards scalable benchmarking of a quantum information processor, it is desirable in many cases to benchmark individual gates in Clifford group rather than the entire set. Interleaving randomised benchmarking is a protocol which consists of interleaving random gates between the gate of interest, which is used to estimate the average error of individual quantum computational gates.

To benchmark a specific Clifford element (an individual gate), the following steps are involved:

'''Step 1''': Implement [[Standard Randomised Benchmarking]] to get a model for the fidelity and to calculate the average error rate

* A fixed sequence length is selected at random. A random sequence of this length is chosen from the Clifford group.
* The operations are applied to the initial state corresponding to the selected sequence and then a final operator is applied which inverts all the previous operations.
* The final state is then measured to check if it matches the initial state. This process is performed several times with the same sequence to estimate the survival probability (the probability that the final state which returns to its initial state).
* Other random sequences of the same fixed sequence length are picked and the above-mentioned process is repeated to calculate the corresponding survival probability. This is then used to calculate the average survival probability for the sequence length.
* The same procedure is repeated for multiple different randomly selected sequence lengths.
* The observed survival probabilities are then plotted against the sequence length and then this is fit to an exponential decay curve, which is used to estimate the depolarizing parameter and sequence fidelity and also to calculate the average error rate which is the metric for randomized benchmarking.

'''Step 2''': Procedure to estimate the new sequence fidelity by including the Clifford element to be benchmarked in the sequence

* Now, for a random fixed sequence length, choose a sequence where the first Clifford element is selected uniformly at random from the Clifford group and the second element is always chosen to be the specific Clifford element we want to benchmark.
* Final gate is chosen to be the inverse of the composition mentioned in the step above. The final state is then measured to check if it matches the initial state. This process is performed several times with the same sequence to estimate the survival probability (the probability that the final state which returns to its initial state).
* Other random sequences of the same fixed sequence length are picked and the above-mentioned process is repeated to calculate the corresponding survival probability. This is then used to calculate the average survival probability for the sequence length.
* The same procedure is repeated for multiple different randomly selected sequence lengths.
* The observed survival probabilities are then plotted against the sequence length, to obtain a zeroth or first-order model of the new sequence fidelity, from which the new depolarizing parameter is estimated.

'''Step 3''': Estimate the gate error of the selected Clifford element to be benchmarked

* From the values obtained for the depolarizing parameter and the new depolarizing parameter, using the average gate fidelity, a point estimate is obtained for the gate error. The gate error would lie in a certain range of this estimate. One interpretation of this error is that it arises from imperfect random gates.

==Hardware Requirements==
* Quantum computational resources to perform Clifford gates.
* Trusted Measurement device.

==Notation==
* <math>p</math>: Depolarizing parameter
* <math>p_{\bar{C}}</math>: New depolarizing parameter for the specific Clifford element to be benchmarked
* <math>d</math>: Dimension of Hilbert space
* <math>F_{avg}</math>: Average fidelity, <math>F_{avg} = p + \frac{1-p}{d}</math>
* <math>r</math>: Average error rate, <math>r = 1- F_{avg}, r = \frac{(d-1)(1-p)}{d}</math>
* <math>m</math>: Selected sequence length
* <math>K_m</math>: Total randomly selected sequence of <math>m</math> sequence length
* Clif<math>_n</math>: Clifford group
* C: Selected Clifford element to be benchmarked
* C<math>_i</math>: Random element of Clifford group
* <math>S_{(i_1, ...,i_m)}</math> = <math>S_{\mathbf{i_m}}</math>: Random sequence of operations of length <math>m</math>
* <math>\gamma</math>: Superoperator representing the sequence with alternating <math>C</math>
* <math>M</math>: Number of different data points to get the error model
* <math>\Lambda_{i,j}</math>: Implementation of C<math>_i</math> at time j (1 <math>\leq</math> j <math>\leq</math> M) results in this error map. <math>\Lambda_{i,1}, ..., \Lambda_{i,M}</math> are the different time-dependent noise operators affecting C<math>_i</math>.
* <math>\Lambda_{C}</math>: Associated noise operator of the Clifford element <math>C</math>
* <math>r^{est}_C</math>: The gate error of <math>\Lambda_{C}</math>
* <math>E</math>: Error range of <math>r^{est}_C</math>
* <math>|\psi\rangle</math>: initial state
* <math>E_{\psi}</math>: POVM element which takes into account the measurement error.
* <math>F_{seq}(m, \psi) = Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>: Survival probability of a sequence. <math>\rho_\psi</math> is a quantum state that takes into account errors in preparing <math>\langle \psi |\psi \rangle</math>
* <math>F_g^{(0)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time independent error model
* <math>F_g^{(1)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time dependent error model. In this model, the parameter <math>(q-p^2)</math> is a measure of the degree of gate-dependence in the error.
* <math>F_\bar{g}^{(0)}(m, |\psi\rangle)</math>: New zeroth order averaged sequence fidelity for <math>C</math>
* <math>F_\bar{g}^{(1)}(m, |\psi\rangle)</math>: New first order Averaged sequence fidelity for <math>C</math>
* <math>A_0, B_0</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time independent error model
* <math>A_1, B_1, C_1</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time dependent error model.
* <math>R_{m+1}</math>: <math>\frac{1}{|Clif_n|}\sum_i\Lambda_{i, m+1} \otimes (C_i \otimes \Lambda \otimes C_i^{\dagger})</math>

==Properties==
* '''Figure of merit''': average gate error

* The errors which are considered here are State preparation and measurement errors, error on the final gate, which are gate and time-independent errors. Gate and time-dependent errors can also be taken into consideration. This method is insensitive to SPAM error.
* The random gates are picked from the Clifford group.
* For noise estimation, the uniform probability distribution over Clifford group comprises a [[unitary 2-design]].
* This protocol provides a scalable method for benchmarking the set of Clifford gates.
* To obtain a more accurate value for <math>p</math> one should always use the first order fitting model unless prior knowledge of the noise indicates that it is effectively gate-independent.

==Procedure Description==

'''Step 1''': Standard Randomised Benchmarking

'''Output''': Figure of merit: <math>r</math>

* For <math>1, 2, ..., M</math>:
** Pick random sequence length <math>m</math>
** For <math>k = 1, 2, ..., K_m</math> sequences:
*** For <math>j = 1, 2 ..., m+1</math>:
**** If <math>j == m+1</math>, apply inverse operator of previous operations
**** else, apply random operation C<math>_i</math>
*** Thus, <math>S_{\mathbf{i_m}} = \bigotimes^{m+1}_{j=1} (\Lambda_{(i_j, j)} \circ C_{i_j})</math> and <math>i_{m+1}</math> is uniquely determined by <math>(i_1, ...,i_m)</math>
*** Measure survival probability <math>Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>
** Estimate average survival probability <math>Tr[E_{\psi}S_{\mathbf{K_m}}(\rho_\psi)]</math> over all <math>K_m</math> sequences, where <math>S_{\mathbf{K_m}} = \frac{1}{K_m}\sum_{i_m} S_{i_m}</math>
* Fit the results for the averaged sequence fidelity for all <math>m</math> into the models:
** For gate and time independent error model:
*** <math>F_g^{(0)}(m, |\psi\rangle) = A_0p^m + B_0</math>
** For gate and time dependent error model:
*** <math>F_g^{(1)}(m, |\psi\rangle) = A_1p^m + B_1 + C_1(m-1)(q-p^2)p^{m-2}</math>
* <math>p</math> is extracted from the model and <math>r</math> is estimated, <math>r = \frac{(d-1)(1-p)}{d}</math>

'''Step 2''': Estimate gate error of selected Clifford element C

'''Input''': C

'''Output''': gate error of <math>\Lambda_{C}</math>: <math>r^{est}_C</math>

* For <math>1, 2, ..., M</math>:
** Pick random sequence length <math>m</math>
** For <math>k = 1, 2, ..., K_m</math> sequences:
*** For <math>j = 1, 2 ..., m+1</math>:
**** If <math>j == m+1</math>, apply inverse operator of previous operations
**** else If <math>j%2==1</math>, apply random operation C<math>_i</math>
**** else, apply C
*** Thus <math>\gamma = \Lambda_{i_{m+1}} +</math> C<math>_{i_{m+1}} (\bigotimes^{m+1}_{j=1}[C \circ \Lambda_C \circ \Lambda_{i_j} \circ C_{i_j}])</math>
*** Measure the survival probability <math>Tr[E_{\psi}\gamma_{\mathbf{i_m}}(\rho_\psi)]</math>
** Estimate average survival probability <math>Tr[E_{\psi}\gamma_{\mathbf{K_m}}(\rho_\psi)]</math> over all <math>K_m</math> sequences, where <math>\gamma_{\mathbf{K_m}} = \frac{1}{K_m}\sum_{i_m} \gamma_{i_m}</math>
* Fit the results for the averaged sequence fidelity for all <math>m</math> into the models, to find <math>p_{\bar{C}}</math>:
** For gate and time independent error model:
*** <math>F_\bar{g}^{(0)}(m, |\psi\rangle) = A_0p_{\bar{C}}^m + B_0</math>
** For gate and time dependent error model:
*** <math>F_\bar{g}^{(1)}(m, |\psi\rangle) = A_1p_{\bar{C}}^m + B_1 + C_1(m-1)(q-p_{\bar{C}}^2)p_{\bar{C}}^{m-2}</math>
* Estimate <math>r^{est}_C = \frac{(d-1)(1-p_{\bar{C}}/p)}{d}</math>
* <math>r^{est}_C</math> lies in the range <math>[r^{est}_C-E, r^{est}_C+E]</math>, where <math>E = min (\frac{(d-1)[|p-p_{\bar{C}}/p| + (1-p)]}{d}, \frac{2(d^2-1)(1-p)}{pd^2} + \frac{4\sqrt{1-p}\sqrt{d^2-1}}{p})</math>

==Further Information==
* Fitting models are described and derived as seen in [https://arxiv.org/abs/1109.6887v2 E. Mageson et al]. The coefficients derived are:
** <math>A_0</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
** <math>B_0</math> = Tr<math>[E_\psi \Lambda(\frac{\mathbb{1}}{d})]</math>
** <math>A_1</math> = Tr<math>[E_\psi \Lambda(\frac{Q_1\rho_\psi}{p} - \rho_\psi + \frac{(p-1)\mathbb{1}}{pd})]</math> + Tr<math>[E_\psi R_{m+1}(\frac{\rho_\psi}{p} - \frac{\mathbb{1}}{pd})]</math>
** <math>B_1</math> = Tr<math>[E_\psi R_{m+1}(\frac{\mathbb{1}}{d})]</math>
** <math>C_1</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
* The case where Randomized benchmarking fails: Suppose the noise is time dependent and for each <math>i, \Lambda_i = C_i^{\dagger}</math>. Then <math>F_g(m, \psi) = 1</math> for every <math>m</math> even though there is a substantial error on each <math>C_i</math> and so benchmarking fails.
* Wallman, Granade, Harper, F., NJP 2015 [[Purity benchmarking]]: A unitarity can be estimated via purity benchmarking, which is an RB-like experiment that estimates a decay rate.

==Related Papers==
* E.Knill et al (2007) arXiv:0707.0963: gate and time-independent noise model
* E. Mageson et al (2011) arXiv:1009.3639: multi-parameter model
* Magesan et al. PRL (2012): Interleaved Randomized Benchmarking

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Interleaved Randomised Benchmarking

2020-03-28T10:18:01Z

Rhea: /* Outline */

[https://arxiv.org/abs/1203.4550 Interleaved Randomized benchmarking] is a scalable experimental protocol for estimating the average error of individual quantum computational gates. This protocol consists of interleaving random Clifford gates between the gate of interest and provides an estimate as well as theoretical bounds for the average error of the gate under test, so long as the average noise variation over all Clifford gates is small. This technique takes into account both state preparation and measurement errors and is scalable in the number of qubits.

'''Tags:''' [[:Category: Certification protocol|Certification Protocol]], [[:Category: Average gate fidelity|Average gate fidelity]], [[Randomised Benchmarking]], Clifford group

==Assumptions==
* The measurements performed are trusted.
* Noise model can be assumed to be gate and time-dependent or gate and time-independent.
* The noise model is independent and identically distributed (IID).

==Outline==
[[Standard Randomised Benchmarking]] method involves applying many random sequences of gates of varying lengths to a standard initial state. Each sequence ends with a randomized measurement that determines whether the correct final state was obtained. The average computationally relevant error per gate is obtained from the increase in error probability of the final measurements as a function of sequence length. The random gates are taken from the [[Clifford group]]. The restriction to the Clifford group ensures that the measurements can be of one-qubit Pauli operators that yield at least one deterministic one-bit answer in the absence of errors.

The multi-qubit RB protocol described in Standard Randomised Benchmarking is restricted to benchmark only the full [[Clifford group]] on <math>n</math> qubits. While this provides a significant step towards scalable benchmarking of a quantum information processor, it is desirable in many cases to benchmark individual gates in Clifford group rather than the entire set. Interleaving randomised benchmarking is a protocol which consists of interleaving random gates between the gate of interest, which is used to estimate the average error of individual quantum computational gates.

To benchmark a specific Clifford element (an individual gate), the following steps are involved:

'''Step 1''': Implement [[Standard Randomised Benchmarking]] to get a model for the fidelity and to calculate the average error rate

* A fixed sequence length is selected at random. A random sequence of this length is chosen from the Clifford group.
* The operations are applied to the initial state corresponding to the selected sequence and then a final operator is applied which inverts all the previous operations.
* The final state is then measured to check if it matches the initial state. This process is performed several times with the same sequence to estimate the survival probability (the probability that the final state which returns to its initial state).
* Other random sequences of the same fixed sequence length are picked and the above-mentioned process is repeated to calculate the corresponding survival probability. This is then used to calculate the average survival probability for the sequence length.
* The same procedure is repeated for multiple different randomly selected sequence lengths.
* The observed survival probabilities are then plotted against the sequence length and then this is fit to an exponential decay curve, which is used to estimate the depolarizing parameter and sequence fidelity and also to calculate the average error rate which is the metric for randomized benchmarking.

'''Step 2''': Procedure to estimate the new sequence fidelity by including the Clifford element to be benchmarked in the sequence

* Now, for a random fixed sequence length, choose a sequence where the first Clifford element is selected uniformly at random from the Clifford group and the second element is always chosen to be the specific Clifford element we want to benchmark.
* Final gate is chosen to be the inverse of the composition mentioned in the step above. The final state is then measured to check if it matches the initial state. This process is performed several times with the same sequence to estimate the survival probability (the probability that the final state which returns to its initial state).
* Other random sequences of the same fixed sequence length are picked and the above-mentioned process is repeated to calculate the corresponding survival probability. This is then used to calculate the average survival probability for the sequence length.
* The same procedure is repeated for multiple different randomly selected sequence lengths.
* The observed survival probabilities are then plotted against the sequence length, to obtain a zeroth or first-order model of the new sequence fidelity, from which the new depolarizing parameter is estimated.

'''Step 3''': Estimate the gate error of the selected Clifford element to be benchmarked

* From the values obtained for the depolarizing parameter and the new depolarizing parameter, using the average gate fidelity, a point estimate is obtained for the gate error. The gate error would lie in a certain range of this estimate.

==Hardware Requirements==
* Quantum computational resources to perform Clifford gates.
* Trusted Measurement device.

==Notation==
* <math>p</math>: Depolarizing parameter
* <math>d</math>: Dimension of Hilbert space
* <math>F_{avg}</math>: Average fidelity, <math>F_{avg} = p + \frac{1-p}{d}</math>
* <math>r</math>: Average error rate, <math>r = 1- F_{avg}, r = \frac{(d-1)(1-p)}{d}</math>
* <math>m</math>: Selected sequence length
* <math>K_m</math>: Total randomly selected sequence of <math>m</math> sequence length
* Clif<math>_n</math>: Clifford group
* C<math>_i</math>: Random element of Clifford group
* <math>S_{(i_1, ...,i_m)}</math> = <math>S_{\mathbf{i_m}}</math>: Random sequence of operations of length <math>m</math>
* <math>M</math>: Number of different data points to get the error model
* <math>\Lambda_{i,j}</math>: Implementation of C<math>_i</math> at time j (1 <math>\leq</math> j <math>\leq</math> M) results in this error map. <math>\Lambda_{i,1}, ..., \Lambda_{i,M}</math> are the different time-dependent noise operators affecting C<math>_i</math>.
* <math>|\psi\rangle</math>: initial state
* <math>E_{\psi}</math>: POVM element which takes into account the measurement error.
* <math>F_{seq}(m, \psi) = Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>: Survival probability of a sequence. <math>\rho_\psi</math> is a quantum state that takes into account errors in preparing <math>\langle \psi |\psi \rangle</math>
* <math>F_g^{(0)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time independent error model
* <math>F_g^{(1)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time dependent error model. In this model, the parameter <math>(q-p^2)</math> is a measure of the degree of gate-dependence in the error.
* <math>A_0, B_0</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time independent error model
* <math>A_1, B_1, C_1</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time dependent error model.
* <math>R_{m+1}</math>: <math>\frac{1}{|Clif_n|}\sum_i\Lambda_{i, m+1} \otimes (C_i \otimes \Lambda \otimes C_i^{\dagger})</math>

==Properties==
* '''Figure of merit''': average error rate, average gate fidelity
* The errors which are considered here are State preparation and measurement errors, error on the final gate, which are gate and time-independent errors. Gate and time-dependent errors can also be taken into consideration. This method is insensitive to SPAM error.
* The random gates are picked from the Clifford group.
* For noise estimation, the uniform probability distribution over Clifford group comprises a [[unitary 2-design]].
* This protocol provides a scalable method for benchmarking the set of Clifford gates.
* To obtain a more accurate value for <math>p</math> one should always use the first order fitting model unless prior knowledge of the noise indicates that it is effectively gate-independent.

==Procedure Description==
'''Output''': Figure of merit: <math>r</math>

* For <math>1, 2, ..., M</math>:
** Pick random sequence length <math>m</math>
** For <math>k = 1, 2, ..., K_m</math> sequences:
*** For <math>j = 1, 2 ..., m+1</math>:
**** If <math>j == m+1</math>, apply inverse operator of previous operations
**** else, apply random operation C<math>_i</math>
*** Thus, <math>S_{\mathbf{i_m}} = \bigotimes^{m+1}_{j=1} (\Lambda_{(i_j, j)} C_{i_j})</math> and <math>i_{m+1}</math> is uniquely determined by <math>(i_1, ...,i_m)</math>
*** Measure survival probability <math>Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>
** Estimate average survival probability <math>Tr[E_{\psi}S_{\mathbf{K_m}}(\rho_\psi)]</math> over all <math>K_m</math> sequences, where <math>S_{\mathbf{K_m}} = \frac{1}{K_m}\sum_{i_m} S_{i_m}</math>
* Fit the results for the averaged sequence fidelity for all <math>m</math> into the models:
** For gate and time independent error model:
*** <math>F_g^{(0)}(m, |\psi\rangle) = A_0p^m + B_0</math>
** For gate and time dependent error model:
*** <math>F_g^{(1)}(m, |\psi\rangle) = A_1p^m + B_1 + C_1(m-1)(q-p^2)p^{m-2}</math>
* <math>p</math> is extracted from the model and <math>r</math> is estimated, <math>r = \frac{(d-1)(1-p)}{d}</math>

==Further Information==
* Fitting models are described and derived as seen in [https://arxiv.org/abs/1109.6887v2 E. Mageson et al]. The coefficients derived are:
** <math>A_0</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
** <math>B_0</math> = Tr<math>[E_\psi \Lambda(\frac{\mathbb{1}}{d})]</math>
** <math>A_1</math> = Tr<math>[E_\psi \Lambda(\frac{Q_1\rho_\psi}{p} - \rho_\psi + \frac{(p-1)\mathbb{1}}{pd})]</math> + Tr<math>[E_\psi R_{m+1}(\frac{\rho_\psi}{p} - \frac{\mathbb{1}}{pd})]</math>
** <math>B_1</math> = Tr<math>[E_\psi R_{m+1}(\frac{\mathbb{1}}{d})]</math>
** <math>C_1</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
* The case where Randomized benchmarking fails: Suppose the noise is time dependent and for each <math>i, \Lambda_i = C_i^{\dagger}</math>. Then <math>F_g(m, \psi) = 1</math> for every <math>m</math> even though there is a substantial error on each <math>C_i</math> and so benchmarking fails.
* [https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.109.080505 Interleaved Randomized Benchmarking]: This protocol consists of interleaving random Clifford gates between the gate of interest and provides an estimate as well as theoretical bounds for the average error of the gate under test, so long as the average noise variation over all Clifford gates is small. Here the procedure followed is:
** Choose <math>K</math> sequences of Clifford elements where the first Clifford <math>C_{i_1}</math> in each sequence is chosen uniformly at random from Clif<math>_n</math>, the second is always chosen to be <math>C</math>(gate of interest), and alternate between uniformly random Clifford elements and deterministic <math>C</math> up to the <math>m^{th}</math> random gate.
** The <math>(m+1)^{th}</math> gate is chosen to be the inverse of the composition of the first <math>m</math> random gates and interlaced <math>C</math> gates.
** The rest of the steps remain the same and finally after plotting the new average sequence fidelity with the sequence length and fitting it into either the gate and time dependent or the gate and time independent model, we receive the new depolarizing parameter obtained is <math>p_c</math>, which replaces <math>p</math>.
** The new gate error is calculated as <math>r_c = \frac{(d-1)(1-p_c/p)}{d}</math>
* Wallman, Granade, Harper, F., NJP 2015 [[Purity benchmarking]]: A unitarity can be estimated via purity benchmarking, which is an RB-like experiment that estimates a decay rate.

==Related Papers==
* E.Knill et al (2007) arXiv:0707.0963: gate and time-independent noise model
* E. Mageson et al (2011) arXiv:1009.3639: multi-parameter model
* Magesan et al. PRL (2012): Interleaved Randomized Benchmarking
* Harper et al (2016) arXiv:1608.02943v2: Interleaved Randomised Benchmarking to estimate fidelity of T gates
* Wallman, Granade, Harper, F., NJP 2015: Purity benchmarking

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Interleaved Randomised Benchmarking

2020-03-27T21:14:33Z

Rhea:

[https://arxiv.org/abs/1203.4550 Interleaved Randomized benchmarking] is a scalable experimental protocol for estimating the average error of individual quantum computational gates. This protocol consists of interleaving random Clifford gates between the gate of interest and provides an estimate as well as theoretical bounds for the average error of the gate under test, so long as the average noise variation over all Clifford gates is small. This technique takes into account both state preparation and measurement errors and is scalable in the number of qubits.

'''Tags:''' [[:Category: Certification protocol|Certification Protocol]], [[:Category: Average gate fidelity|Average gate fidelity]], [[Randomised Benchmarking]], Clifford group

==Assumptions==
* The measurements performed are trusted.
* Noise model can be assumed to be gate and time-dependent or gate and time-independent.
* The noise model is independent and identically distributed (IID).

==Outline==
[[Standard Randomised Benchmarking]] method involves applying many random sequences of gates of varying lengths to a standard initial state. Each sequence ends with a randomized measurement that determines whether the correct final state was obtained. The average computationally relevant error per gate is obtained from the increase in error probability of the final measurements as a function of sequence length. The random gates are taken from the [[Clifford group]]. The restriction to the Clifford group ensures that the measurements can be of one-qubit Pauli operators that yield at least one deterministic one-bit answer in the absence of errors.

The multi-qubit RB protocol described in Standard Randomised Benchmarking is restricted to benchmark only the full [[Clifford group]] on <math>n</math> qubits. While this provides a significant step towards scalable benchmarking of a quantum information processor, it is desirable in many cases to benchmark individual gates in Clifford group rather than the entire set. Interleaving randomised benchmarking is a protocol which consists of interleaving random gates between the gate of interest, which is used to estimate the average error of individual quantum computational gates.

To benchmark a specific Clifford element (an individual gate), the following steps are involved:

'''Step 1''': Implement [[Standard Randomised Benchmarking]] to get a model for the fidelity and to calculate the average error rate

* A fixed sequence length is selected at random. A random sequence of this length is chosen from the Clifford group.
* The operations are applied to the initial state corresponding to the selected sequence and then a final operator is applied which inverts all the previous operations.
* The final state is then measured to check if it matches the initial state. This process is performed several times with the same sequence to estimate the survival probability (the probability that the final state which returns to its initial state).
* Other random sequences of the same fixed sequence length are picked and the above-mentioned process is repeated to calculate the corresponding survival probability. This is then used to calculate the average survival probability for the sequence length.
* The same procedure is repeated for multiple different randomly selected sequence lengths.
* The observed survival probabilities are then plotted against the sequence length and then this is fit to an exponential decay curve, which is used to estimate the depolarizing parameter and sequence fidelity and also to calculate the average error rate which is the metric for randomized benchmarking.

'''Step 2''': Procedure to estimate the new sequence fidelity by including the Clifford element to be benchmarked in the sequence

* Now, for a random fixed sequence length, choose a sequence where the first Clifford element is selected uniformly at random from the Clifford group and the second element is always chosen to be the specific Clifford element we want to benchmark.
* Final gate is chosen to be the inverse of the composition mentioned in the step above. The final state is then measured to check if it matches the initial state. This process is performed several times with the same sequence to estimate the survival probability (the probability that the final state which returns to its initial state).
* Other random sequences of the same fixed sequence length are picked and the above-mentioned process is repeated to calculate the corresponding survival probability. This is then used to calculate the average survival probability for the sequence length.
* The same procedure is repeated for multiple different randomly selected sequence lengths.
* The observed survival probabilities are then plotted against the sequence length, to obtain a zeroth or first-order model of the new sequence fidelity, from which the new depolarizing parameter is estimated.

==Hardware Requirements==
* Quantum computational resources to perform Clifford gates.
* Trusted Measurement device.

==Notation==
* <math>p</math>: Depolarizing parameter
* <math>d</math>: Dimension of Hilbert space
* <math>F_{avg}</math>: Average fidelity, <math>F_{avg} = p + \frac{1-p}{d}</math>
* <math>r</math>: Average error rate, <math>r = 1- F_{avg}, r = \frac{(d-1)(1-p)}{d}</math>
* <math>m</math>: Selected sequence length
* <math>K_m</math>: Total randomly selected sequence of <math>m</math> sequence length
* Clif<math>_n</math>: Clifford group
* C<math>_i</math>: Random element of Clifford group
* <math>S_{(i_1, ...,i_m)}</math> = <math>S_{\mathbf{i_m}}</math>: Random sequence of operations of length <math>m</math>
* <math>M</math>: Number of different data points to get the error model
* <math>\Lambda_{i,j}</math>: Implementation of C<math>_i</math> at time j (1 <math>\leq</math> j <math>\leq</math> M) results in this error map. <math>\Lambda_{i,1}, ..., \Lambda_{i,M}</math> are the different time-dependent noise operators affecting C<math>_i</math>.
* <math>|\psi\rangle</math>: initial state
* <math>E_{\psi}</math>: POVM element which takes into account the measurement error.
* <math>F_{seq}(m, \psi) = Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>: Survival probability of a sequence. <math>\rho_\psi</math> is a quantum state that takes into account errors in preparing <math>\langle \psi |\psi \rangle</math>
* <math>F_g^{(0)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time independent error model
* <math>F_g^{(1)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time dependent error model. In this model, the parameter <math>(q-p^2)</math> is a measure of the degree of gate-dependence in the error.
* <math>A_0, B_0</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time independent error model
* <math>A_1, B_1, C_1</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time dependent error model.
* <math>R_{m+1}</math>: <math>\frac{1}{|Clif_n|}\sum_i\Lambda_{i, m+1} \otimes (C_i \otimes \Lambda \otimes C_i^{\dagger})</math>

==Properties==
* '''Figure of merit''': average error rate, average gate fidelity
* The errors which are considered here are State preparation and measurement errors, error on the final gate, which are gate and time-independent errors. Gate and time-dependent errors can also be taken into consideration. This method is insensitive to SPAM error.
* The random gates are picked from the Clifford group.
* For noise estimation, the uniform probability distribution over Clifford group comprises a [[unitary 2-design]].
* This protocol provides a scalable method for benchmarking the set of Clifford gates.
* To obtain a more accurate value for <math>p</math> one should always use the first order fitting model unless prior knowledge of the noise indicates that it is effectively gate-independent.

==Procedure Description==
'''Output''': Figure of merit: <math>r</math>

* For <math>1, 2, ..., M</math>:
** Pick random sequence length <math>m</math>
** For <math>k = 1, 2, ..., K_m</math> sequences:
*** For <math>j = 1, 2 ..., m+1</math>:
**** If <math>j == m+1</math>, apply inverse operator of previous operations
**** else, apply random operation C<math>_i</math>
*** Thus, <math>S_{\mathbf{i_m}} = \bigotimes^{m+1}_{j=1} (\Lambda_{(i_j, j)} C_{i_j})</math> and <math>i_{m+1}</math> is uniquely determined by <math>(i_1, ...,i_m)</math>
*** Measure survival probability <math>Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>
** Estimate average survival probability <math>Tr[E_{\psi}S_{\mathbf{K_m}}(\rho_\psi)]</math> over all <math>K_m</math> sequences, where <math>S_{\mathbf{K_m}} = \frac{1}{K_m}\sum_{i_m} S_{i_m}</math>
* Fit the results for the averaged sequence fidelity for all <math>m</math> into the models:
** For gate and time independent error model:
*** <math>F_g^{(0)}(m, |\psi\rangle) = A_0p^m + B_0</math>
** For gate and time dependent error model:
*** <math>F_g^{(1)}(m, |\psi\rangle) = A_1p^m + B_1 + C_1(m-1)(q-p^2)p^{m-2}</math>
* <math>p</math> is extracted from the model and <math>r</math> is estimated, <math>r = \frac{(d-1)(1-p)}{d}</math>

==Further Information==
* Fitting models are described and derived as seen in [https://arxiv.org/abs/1109.6887v2 E. Mageson et al]. The coefficients derived are:
** <math>A_0</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
** <math>B_0</math> = Tr<math>[E_\psi \Lambda(\frac{\mathbb{1}}{d})]</math>
** <math>A_1</math> = Tr<math>[E_\psi \Lambda(\frac{Q_1\rho_\psi}{p} - \rho_\psi + \frac{(p-1)\mathbb{1}}{pd})]</math> + Tr<math>[E_\psi R_{m+1}(\frac{\rho_\psi}{p} - \frac{\mathbb{1}}{pd})]</math>
** <math>B_1</math> = Tr<math>[E_\psi R_{m+1}(\frac{\mathbb{1}}{d})]</math>
** <math>C_1</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
* The case where Randomized benchmarking fails: Suppose the noise is time dependent and for each <math>i, \Lambda_i = C_i^{\dagger}</math>. Then <math>F_g(m, \psi) = 1</math> for every <math>m</math> even though there is a substantial error on each <math>C_i</math> and so benchmarking fails.
* [https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.109.080505 Interleaved Randomized Benchmarking]: This protocol consists of interleaving random Clifford gates between the gate of interest and provides an estimate as well as theoretical bounds for the average error of the gate under test, so long as the average noise variation over all Clifford gates is small. Here the procedure followed is:
** Choose <math>K</math> sequences of Clifford elements where the first Clifford <math>C_{i_1}</math> in each sequence is chosen uniformly at random from Clif<math>_n</math>, the second is always chosen to be <math>C</math>(gate of interest), and alternate between uniformly random Clifford elements and deterministic <math>C</math> up to the <math>m^{th}</math> random gate.
** The <math>(m+1)^{th}</math> gate is chosen to be the inverse of the composition of the first <math>m</math> random gates and interlaced <math>C</math> gates.
** The rest of the steps remain the same and finally after plotting the new average sequence fidelity with the sequence length and fitting it into either the gate and time dependent or the gate and time independent model, we receive the new depolarizing parameter obtained is <math>p_c</math>, which replaces <math>p</math>.
** The new gate error is calculated as <math>r_c = \frac{(d-1)(1-p_c/p)}{d}</math>
* Wallman, Granade, Harper, F., NJP 2015 [[Purity benchmarking]]: A unitarity can be estimated via purity benchmarking, which is an RB-like experiment that estimates a decay rate.

==Related Papers==
* E.Knill et al (2007) arXiv:0707.0963: gate and time-independent noise model
* E. Mageson et al (2011) arXiv:1009.3639: multi-parameter model
* Magesan et al. PRL (2012): Interleaved Randomized Benchmarking
* Harper et al (2016) arXiv:1608.02943v2: Interleaved Randomised Benchmarking to estimate fidelity of T gates
* Wallman, Granade, Harper, F., NJP 2015: Purity benchmarking

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Interleaved Randomised Benchmarking

2020-03-27T13:57:28Z

Rhea: Created page with "[https://arxiv.org/abs/1109.6887v2 Interleaved Randomized benchmarking] is a scalable experimental protocol for estimating the average error of individual quantum computationa..."

[https://arxiv.org/abs/1109.6887v2 Interleaved Randomized benchmarking] is a scalable experimental protocol for estimating the average error of individual quantum computational gates. This protocol consists of interleaving random Clifford gates between the gate of interest and provides an estimate as well as theoretical bounds for the average error of the gate under test, so long as the average noise variation over all Clifford gates is small. This technique takes into account both state preparation and measurement errors and is scalable in the number of qubits.

'''Tags:''' [[:Category: Certification protocol|Certification Protocol]], [[:Category: Average gate fidelity|Average gate fidelity]], [[Randomised Benchmarking]], Clifford group

==Assumptions==
* The measurements performed are trusted.
* Noise model can be assumed to be gate and time-dependent or gate and time-independent.
* The noise model is independent and identically distributed (IID).

==Outline==
Randomized benchmarking method involves applying many random sequences of gates of varying lengths to a standard initial state. Each sequence ends with a randomized measurement that determines whether the correct final state was obtained. The average computationally relevant error per gate is obtained from the increase in error probability of the final measurements as a function of sequence length.

The random gates are taken from the [[Clifford group]]. The restriction to the Clifford group ensures that the measurements can be of one-qubit Pauli operators that yield at least one deterministic one-bit answer in the absence of errors.

This method consists of the following steps:

* A fixed sequence length is selected at random. A random sequence of this length is chosen from the Clifford group.
* The operations are applied to the initial state corresponding to the selected sequence and then a final operator is applied which inverts all the previous operations.
* The final state is then measured to check if it matches the initial state. This process is performed several times with the same sequence to estimate the survival probability (the probability that the final state which returns to its initial state).
* Other random sequences of the same fixed sequence length are picked and the above-mentioned process is repeated to calculate the corresponding survival probability. This is then used to calculate the average survival probability for the sequence length.
* The same procedure is repeated for multiple different randomly selected sequence lengths.
* The observed survival probabilities are then plotted against the sequence length and then this is fit to an exponential decay curve, which is used to estimate the fidelity and also to calculate the average error rate which is the metric for randomized benchmarking.

==Hardware Requirements==
* Quantum computational resources to perform Clifford gates.
* Trusted Measurement device.

==Notation==
* <math>p</math>: Depolarizing parameter
* <math>d</math>: Dimension of Hilbert space
* <math>F_{avg}</math>: Average fidelity, <math>F_{avg} = p + \frac{1-p}{d}</math>
* <math>r</math>: Average error rate, <math>r = 1- F_{avg}, r = \frac{(d-1)(1-p)}{d}</math>
* <math>m</math>: Selected sequence length
* <math>K_m</math>: Total randomly selected sequence of <math>m</math> sequence length
* Clif<math>_n</math>: Clifford group
* C<math>_i</math>: Random element of Clifford group
* <math>S_{(i_1, ...,i_m)}</math> = <math>S_{\mathbf{i_m}}</math>: Random sequence of operations of length <math>m</math>
* <math>M</math>: Number of different data points to get the error model
* <math>\Lambda_{i,j}</math>: Implementation of C<math>_i</math> at time j (1 <math>\leq</math> j <math>\leq</math> M) results in this error map. <math>\Lambda_{i,1}, ..., \Lambda_{i,M}</math> are the different time-dependent noise operators affecting C<math>_i</math>.
* <math>|\psi\rangle</math>: initial state
* <math>E_{\psi}</math>: POVM element which takes into account the measurement error.
* <math>F_{seq}(m, \psi) = Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>: Survival probability of a sequence. <math>\rho_\psi</math> is a quantum state that takes into account errors in preparing <math>\langle \psi |\psi \rangle</math>
* <math>F_g^{(0)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time independent error model
* <math>F_g^{(1)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time dependent error model. In this model, the parameter <math>(q-p^2)</math> is a measure of the degree of gate-dependence in the error.
* <math>A_0, B_0</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time independent error model
* <math>A_1, B_1, C_1</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time dependent error model.
* <math>R_{m+1}</math>: <math>\frac{1}{|Clif_n|}\sum_i\Lambda_{i, m+1} \otimes (C_i \otimes \Lambda \otimes C_i^{\dagger})</math>

==Properties==
* '''Figure of merit''': average error rate, average gate fidelity
* The errors which are considered here are State preparation and measurement errors, error on the final gate, which are gate and time-independent errors. Gate and time-dependent errors can also be taken into consideration. This method is insensitive to SPAM error.
* The random gates are picked from the Clifford group.
* For noise estimation, the uniform probability distribution over Clifford group comprises a [[unitary 2-design]].
* This protocol provides a scalable method for benchmarking the set of Clifford gates.
* To obtain a more accurate value for <math>p</math> one should always use the first order fitting model unless prior knowledge of the noise indicates that it is effectively gate-independent.

==Procedure Description==
'''Output''': Figure of merit: <math>r</math>

* For <math>1, 2, ..., M</math>:
** Pick random sequence length <math>m</math>
** For <math>k = 1, 2, ..., K_m</math> sequences:
*** For <math>j = 1, 2 ..., m+1</math>:
**** If <math>j == m+1</math>, apply inverse operator of previous operations
**** else, apply random operation C<math>_i</math>
*** Thus, <math>S_{\mathbf{i_m}} = \bigotimes^{m+1}_{j=1} (\Lambda_{(i_j, j)} C_{i_j})</math> and <math>i_{m+1}</math> is uniquely determined by <math>(i_1, ...,i_m)</math>
*** Measure survival probability <math>Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>
** Estimate average survival probability <math>Tr[E_{\psi}S_{\mathbf{K_m}}(\rho_\psi)]</math> over all <math>K_m</math> sequences, where <math>S_{\mathbf{K_m}} = \frac{1}{K_m}\sum_{i_m} S_{i_m}</math>
* Fit the results for the averaged sequence fidelity for all <math>m</math> into the models:
** For gate and time independent error model:
*** <math>F_g^{(0)}(m, |\psi\rangle) = A_0p^m + B_0</math>
** For gate and time dependent error model:
*** <math>F_g^{(1)}(m, |\psi\rangle) = A_1p^m + B_1 + C_1(m-1)(q-p^2)p^{m-2}</math>
* <math>p</math> is extracted from the model and <math>r</math> is estimated, <math>r = \frac{(d-1)(1-p)}{d}</math>

==Further Information==
* Fitting models are described and derived as seen in [https://arxiv.org/abs/1109.6887v2 E. Mageson et al]. The coefficients derived are:
** <math>A_0</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
** <math>B_0</math> = Tr<math>[E_\psi \Lambda(\frac{\mathbb{1}}{d})]</math>
** <math>A_1</math> = Tr<math>[E_\psi \Lambda(\frac{Q_1\rho_\psi}{p} - \rho_\psi + \frac{(p-1)\mathbb{1}}{pd})]</math> + Tr<math>[E_\psi R_{m+1}(\frac{\rho_\psi}{p} - \frac{\mathbb{1}}{pd})]</math>
** <math>B_1</math> = Tr<math>[E_\psi R_{m+1}(\frac{\mathbb{1}}{d})]</math>
** <math>C_1</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
* The case where Randomized benchmarking fails: Suppose the noise is time dependent and for each <math>i, \Lambda_i = C_i^{\dagger}</math>. Then <math>F_g(m, \psi) = 1</math> for every <math>m</math> even though there is a substantial error on each <math>C_i</math> and so benchmarking fails.
* [https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.109.080505 Interleaved Randomized Benchmarking]: This protocol consists of interleaving random Clifford gates between the gate of interest and provides an estimate as well as theoretical bounds for the average error of the gate under test, so long as the average noise variation over all Clifford gates is small. Here the procedure followed is:
** Choose <math>K</math> sequences of Clifford elements where the first Clifford <math>C_{i_1}</math> in each sequence is chosen uniformly at random from Clif<math>_n</math>, the second is always chosen to be <math>C</math>(gate of interest), and alternate between uniformly random Clifford elements and deterministic <math>C</math> up to the <math>m^{th}</math> random gate.
** The <math>(m+1)^{th}</math> gate is chosen to be the inverse of the composition of the first <math>m</math> random gates and interlaced <math>C</math> gates.
** The rest of the steps remain the same and finally after plotting the new average sequence fidelity with the sequence length and fitting it into either the gate and time dependent or the gate and time independent model, we receive the new depolarizing parameter obtained is <math>p_c</math>, which replaces <math>p</math>.
** The new gate error is calculated as <math>r_c = \frac{(d-1)(1-p_c/p)}{d}</math>
* Wallman, Granade, Harper, F., NJP 2015 [[Purity benchmarking]]: A unitarity can be estimated via purity benchmarking, which is an RB-like experiment that estimates a decay rate.

==Related Papers==
* E.Knill et al (2007) arXiv:0707.0963: gate and time-independent noise model
* E. Mageson et al (2011) arXiv:1009.3639: multi-parameter model
* Magesan et al. PRL (2012): Interleaved Randomized Benchmarking
* Harper et al (2016) arXiv:1608.02943v2: Interleaved Randomised Benchmarking to estimate fidelity of T gates
* Wallman, Granade, Harper, F., NJP 2015: Purity benchmarking

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Standard Randomised Benchmarking

2020-03-27T13:55:53Z

Rhea: /* Outline */

[https://arxiv.org/abs/1109.6887v2 Randomized benchmarking] is a protocol that yields estimates of the computationally relevant errors without relying on accurate quantum state preparation and measurement. This is used to determine the error probability per gate in computational context and also gives an overall [[average fidelity]] for the noise in the gates.

'''Tags:''' [[:Category: Certification protocol|Certification Protocol]], [[:Category: Average gate fidelity|Average gate fidelity]], [[Randomised Benchmarking]], Clifford group

==Assumptions==
* The measurements performed are trusted.
* Noise model can be assumed to be gate and time-dependent or gate and time-independent.
* The noise model is independent and identically distributed (IID).

==Outline==
Randomized benchmarking method involves applying many random sequences of gates of varying lengths to a standard initial state. Each sequence ends with a randomized measurement that determines whether the correct final state was obtained. The average computationally relevant error per gate is obtained from the increase in error probability of the final measurements as a function of sequence length.

The random gates are taken from the [[Clifford group]]. The restriction to the Clifford group ensures that the measurements can be of one-qubit Pauli operators that yield at least one deterministic one-bit answer in the absence of errors.

This method consists of the following steps:

* A fixed sequence length is selected at random. A random sequence of this length is chosen from the Clifford group.
* The operations are applied to the initial state corresponding to the selected sequence and then a final operator is applied which inverts all the previous operations.
* The final state is then measured to check if it matches the initial state. This process is performed several times with the same sequence to estimate the survival probability (the probability that the final state which returns to its initial state).
* Other random sequences of the same fixed sequence length are picked and the above-mentioned process is repeated to calculate the corresponding survival probability. This is then used to calculate the average survival probability for the sequence length.
* The same procedure is repeated for multiple different randomly selected sequence lengths.
* The observed survival probabilities are then plotted against the sequence length and then this is fit to an exponential decay curve, which is used to estimate the fidelity and also to calculate the average error rate which is the metric for randomized benchmarking.

==Hardware Requirements==
* Quantum computational resources to perform Clifford gates.
* Trusted Measurement device.

==Notation==
* <math>p</math>: Depolarizing parameter
* <math>d</math>: Dimension of Hilbert space
* <math>F_{avg}</math>: Average fidelity, <math>F_{avg} = p + \frac{1-p}{d}</math>
* <math>r</math>: Average error rate, <math>r = 1- F_{avg}, r = \frac{(d-1)(1-p)}{d}</math>
* <math>m</math>: Selected sequence length
* <math>K_m</math>: Total randomly selected sequence of <math>m</math> sequence length
* Clif<math>_n</math>: Clifford group
* C<math>_i</math>: Random element of Clifford group
* <math>S_{(i_1, ...,i_m)}</math> = <math>S_{\mathbf{i_m}}</math>: Random sequence of operations of length <math>m</math>
* <math>M</math>: Number of different data points to get the error model
* <math>\Lambda_{i,j}</math>: Implementation of C<math>_i</math> at time j (1 <math>\leq</math> j <math>\leq</math> M) results in this error map. <math>\Lambda_{i,1}, ..., \Lambda_{i,M}</math> are the different time-dependent noise operators affecting C<math>_i</math>.
* <math>|\psi\rangle</math>: initial state
* <math>E_{\psi}</math>: POVM element which takes into account the measurement error.
* <math>F_{seq}(m, \psi) = Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>: Survival probability of a sequence. <math>\rho_\psi</math> is a quantum state that takes into account errors in preparing <math>\langle \psi |\psi \rangle</math>
* <math>F_g^{(0)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time independent error model
* <math>F_g^{(1)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time dependent error model. In this model, the parameter <math>(q-p^2)</math> is a measure of the degree of gate-dependence in the error.
* <math>A_0, B_0</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time independent error model
* <math>A_1, B_1, C_1</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time dependent error model.
* <math>R_{m+1}</math>: <math>\frac{1}{|Clif_n|}\sum_i\Lambda_{i, m+1} \otimes (C_i \otimes \Lambda \otimes C_i^{\dagger})</math>

==Properties==
* '''Figure of merit''': average error rate, average gate fidelity
* The errors which are considered here are State preparation and measurement errors, error on the final gate, which are gate and time-independent errors. Gate and time-dependent errors can also be taken into consideration. This method is insensitive to SPAM error.
* The random gates are picked from the Clifford group.
* For noise estimation, the uniform probability distribution over Clifford group comprises a [[unitary 2-design]].
* This protocol provides a scalable method for benchmarking the set of Clifford gates.
* To obtain a more accurate value for <math>p</math> one should always use the first order fitting model unless prior knowledge of the noise indicates that it is effectively gate-independent.

==Procedure Description==
'''Output''': Figure of merit: <math>r</math>

* For <math>1, 2, ..., M</math>:
** Pick random sequence length <math>m</math>
** For <math>k = 1, 2, ..., K_m</math> sequences:
*** For <math>j = 1, 2 ..., m+1</math>:
**** If <math>j == m+1</math>, apply inverse operator of previous operations
**** else, apply random operation C<math>_i</math>
*** Thus, <math>S_{\mathbf{i_m}} = \bigotimes^{m+1}_{j=1} (\Lambda_{(i_j, j)} C_{i_j})</math> and <math>i_{m+1}</math> is uniquely determined by <math>(i_1, ...,i_m)</math>
*** Measure survival probability <math>Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>
** Estimate average survival probability <math>Tr[E_{\psi}S_{\mathbf{K_m}}(\rho_\psi)]</math> over all <math>K_m</math> sequences, where <math>S_{\mathbf{K_m}} = \frac{1}{K_m}\sum_{i_m} S_{i_m}</math>
* Fit the results for the averaged sequence fidelity for all <math>m</math> into the models:
** For gate and time independent error model:
*** <math>F_g^{(0)}(m, |\psi\rangle) = A_0p^m + B_0</math>
** For gate and time dependent error model:
*** <math>F_g^{(1)}(m, |\psi\rangle) = A_1p^m + B_1 + C_1(m-1)(q-p^2)p^{m-2}</math>
* <math>p</math> is extracted from the model and <math>r</math> is estimated, <math>r = \frac{(d-1)(1-p)}{d}</math>

==Further Information==
* Fitting models are described and derived as seen in [https://arxiv.org/abs/1109.6887v2 E. Mageson et al]. The coefficients derived are:
** <math>A_0</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
** <math>B_0</math> = Tr<math>[E_\psi \Lambda(\frac{\mathbb{1}}{d})]</math>
** <math>A_1</math> = Tr<math>[E_\psi \Lambda(\frac{Q_1\rho_\psi}{p} - \rho_\psi + \frac{(p-1)\mathbb{1}}{pd})]</math> + Tr<math>[E_\psi R_{m+1}(\frac{\rho_\psi}{p} - \frac{\mathbb{1}}{pd})]</math>
** <math>B_1</math> = Tr<math>[E_\psi R_{m+1}(\frac{\mathbb{1}}{d})]</math>
** <math>C_1</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
* The case where Randomized benchmarking fails: Suppose the noise is time dependent and for each <math>i, \Lambda_i = C_i^{\dagger}</math>. Then <math>F_g(m, \psi) = 1</math> for every <math>m</math> even though there is a substantial error on each <math>C_i</math> and so benchmarking fails.
* [https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.109.080505 Interleaved Randomized Benchmarking]: This protocol consists of interleaving random Clifford gates between the gate of interest and provides an estimate as well as theoretical bounds for the average error of the gate under test, so long as the average noise variation over all Clifford gates is small. Here the procedure followed is:
** Choose <math>K</math> sequences of Clifford elements where the first Clifford <math>C_{i_1}</math> in each sequence is chosen uniformly at random from Clif<math>_n</math>, the second is always chosen to be <math>C</math>(gate of interest), and alternate between uniformly random Clifford elements and deterministic <math>C</math> up to the <math>m^{th}</math> random gate.
** The <math>(m+1)^{th}</math> gate is chosen to be the inverse of the composition of the first <math>m</math> random gates and interlaced <math>C</math> gates.
** The rest of the steps remain the same and finally after plotting the new average sequence fidelity with the sequence length and fitting it into either the gate and time dependent or the gate and time independent model, we receive the new depolarizing parameter obtained is <math>p_c</math>, which replaces <math>p</math>.
** The new gate error is calculated as <math>r_c = \frac{(d-1)(1-p_c/p)}{d}</math>
* Wallman, Granade, Harper, F., NJP 2015 [[Purity benchmarking]]: A unitarity can be estimated via purity benchmarking, which is an RB-like experiment that estimates a decay rate.

==Related Papers==
* E.Knill et al (2007) arXiv:0707.0963: gate and time-independent noise model
* E. Mageson et al (2011) arXiv:1009.3639: multi-parameter model
* Magesan et al. PRL (2012): Interleaved Randomized Benchmarking
* Harper et al (2016) arXiv:1608.02943v2: Interleaved Randomised Benchmarking to estimate fidelity of T gates
* Wallman, Granade, Harper, F., NJP 2015: Purity benchmarking

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Standard Randomised Benchmarking

2020-03-27T13:47:43Z

Rhea: /* Outline */

[https://arxiv.org/abs/1109.6887v2 Randomized benchmarking] is a protocol that yields estimates of the computationally relevant errors without relying on accurate quantum state preparation and measurement. This is used to determine the error probability per gate in computational context and also gives an overall [[average fidelity]] for the noise in the gates.

'''Tags:''' [[:Category: Certification protocol|Certification Protocol]], [[:Category: Average gate fidelity|Average gate fidelity]], [[Randomised Benchmarking]], Clifford group

==Assumptions==
* The measurements performed are trusted.
* Noise model can be assumed to be gate and time-dependent or gate and time-independent.
* The noise model is independent and identically distributed (IID).

==Outline==
Randomized benchmarking method involves applying many random sequences of gates of varying lengths to a standard initial state. Each sequence ends with a randomized measurement that determines whether the correct final state was obtained. The average computationally relevant error per gate is obtained from the increase in error probability of the final measurements as a function of sequence length.

The random gates are taken from the [[Clifford group]]. The restriction to the Clifford group ensures that the measurements can be of one-qubit Pauli operators that yield at least one deterministic one-bit answer in the absence of errors.

This method consists of the following steps:

* A fixed sequence length is selected which is smaller than a predefined maximum sequence length. A random sequence of this length is chosen from the Clifford group.
* The operations are applied to the initial state corresponding to the selected sequence and then a final operator is applied which inverts all the previous operations.
* The final state is then measured to check if it matches the initial state. This process is performed several times with the same sequence to estimate the survival probability (the probability that the final state which returns to its initial state).
* Other random sequences of the same fixed sequence length are picked and the above-mentioned process is repeated to calculate the corresponding survival probability. This is used to calculate the average survival probability for the sequence length.
* The same procedure is repeated for multiple different randomly selected sequence lengths.
* The observed survival probabilities are then plotted against the sequence length and then this is fit to an exponential decay curve, which is used to estimate the fidelity and also to calculate the average error rate which is the metric for randomized benchmarking.

==Hardware Requirements==
* Quantum computational resources to perform Clifford gates.
* Trusted Measurement device.

==Notation==
* <math>p</math>: Depolarizing parameter
* <math>d</math>: Dimension of Hilbert space
* <math>F_{avg}</math>: Average fidelity, <math>F_{avg} = p + \frac{1-p}{d}</math>
* <math>r</math>: Average error rate, <math>r = 1- F_{avg}, r = \frac{(d-1)(1-p)}{d}</math>
* <math>m</math>: Selected sequence length
* <math>K_m</math>: Total randomly selected sequence of <math>m</math> sequence length
* Clif<math>_n</math>: Clifford group
* C<math>_i</math>: Random element of Clifford group
* <math>S_{(i_1, ...,i_m)}</math> = <math>S_{\mathbf{i_m}}</math>: Random sequence of operations of length <math>m</math>
* <math>M</math>: Number of different data points to get the error model
* <math>\Lambda_{i,j}</math>: Implementation of C<math>_i</math> at time j (1 <math>\leq</math> j <math>\leq</math> M) results in this error map. <math>\Lambda_{i,1}, ..., \Lambda_{i,M}</math> are the different time-dependent noise operators affecting C<math>_i</math>.
* <math>|\psi\rangle</math>: initial state
* <math>E_{\psi}</math>: POVM element which takes into account the measurement error.
* <math>F_{seq}(m, \psi) = Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>: Survival probability of a sequence. <math>\rho_\psi</math> is a quantum state that takes into account errors in preparing <math>\langle \psi |\psi \rangle</math>
* <math>F_g^{(0)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time independent error model
* <math>F_g^{(1)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time dependent error model. In this model, the parameter <math>(q-p^2)</math> is a measure of the degree of gate-dependence in the error.
* <math>A_0, B_0</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time independent error model
* <math>A_1, B_1, C_1</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time dependent error model.
* <math>R_{m+1}</math>: <math>\frac{1}{|Clif_n|}\sum_i\Lambda_{i, m+1} \otimes (C_i \otimes \Lambda \otimes C_i^{\dagger})</math>

==Properties==
* '''Figure of merit''': average error rate, average gate fidelity
* The errors which are considered here are State preparation and measurement errors, error on the final gate, which are gate and time-independent errors. Gate and time-dependent errors can also be taken into consideration. This method is insensitive to SPAM error.
* The random gates are picked from the Clifford group.
* For noise estimation, the uniform probability distribution over Clifford group comprises a [[unitary 2-design]].
* This protocol provides a scalable method for benchmarking the set of Clifford gates.
* To obtain a more accurate value for <math>p</math> one should always use the first order fitting model unless prior knowledge of the noise indicates that it is effectively gate-independent.

==Procedure Description==
'''Output''': Figure of merit: <math>r</math>

* For <math>1, 2, ..., M</math>:
** Pick random sequence length <math>m</math>
** For <math>k = 1, 2, ..., K_m</math> sequences:
*** For <math>j = 1, 2 ..., m+1</math>:
**** If <math>j == m+1</math>, apply inverse operator of previous operations
**** else, apply random operation C<math>_i</math>
*** Thus, <math>S_{\mathbf{i_m}} = \bigotimes^{m+1}_{j=1} (\Lambda_{(i_j, j)} C_{i_j})</math> and <math>i_{m+1}</math> is uniquely determined by <math>(i_1, ...,i_m)</math>
*** Measure survival probability <math>Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>
** Estimate average survival probability <math>Tr[E_{\psi}S_{\mathbf{K_m}}(\rho_\psi)]</math> over all <math>K_m</math> sequences, where <math>S_{\mathbf{K_m}} = \frac{1}{K_m}\sum_{i_m} S_{i_m}</math>
* Fit the results for the averaged sequence fidelity for all <math>m</math> into the models:
** For gate and time independent error model:
*** <math>F_g^{(0)}(m, |\psi\rangle) = A_0p^m + B_0</math>
** For gate and time dependent error model:
*** <math>F_g^{(1)}(m, |\psi\rangle) = A_1p^m + B_1 + C_1(m-1)(q-p^2)p^{m-2}</math>
* <math>p</math> is extracted from the model and <math>r</math> is estimated, <math>r = \frac{(d-1)(1-p)}{d}</math>

==Further Information==
* Fitting models are described and derived as seen in [https://arxiv.org/abs/1109.6887v2 E. Mageson et al]. The coefficients derived are:
** <math>A_0</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
** <math>B_0</math> = Tr<math>[E_\psi \Lambda(\frac{\mathbb{1}}{d})]</math>
** <math>A_1</math> = Tr<math>[E_\psi \Lambda(\frac{Q_1\rho_\psi}{p} - \rho_\psi + \frac{(p-1)\mathbb{1}}{pd})]</math> + Tr<math>[E_\psi R_{m+1}(\frac{\rho_\psi}{p} - \frac{\mathbb{1}}{pd})]</math>
** <math>B_1</math> = Tr<math>[E_\psi R_{m+1}(\frac{\mathbb{1}}{d})]</math>
** <math>C_1</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
* The case where Randomized benchmarking fails: Suppose the noise is time dependent and for each <math>i, \Lambda_i = C_i^{\dagger}</math>. Then <math>F_g(m, \psi) = 1</math> for every <math>m</math> even though there is a substantial error on each <math>C_i</math> and so benchmarking fails.
* [https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.109.080505 Interleaved Randomized Benchmarking]: This protocol consists of interleaving random Clifford gates between the gate of interest and provides an estimate as well as theoretical bounds for the average error of the gate under test, so long as the average noise variation over all Clifford gates is small. Here the procedure followed is:
** Choose <math>K</math> sequences of Clifford elements where the first Clifford <math>C_{i_1}</math> in each sequence is chosen uniformly at random from Clif<math>_n</math>, the second is always chosen to be <math>C</math>(gate of interest), and alternate between uniformly random Clifford elements and deterministic <math>C</math> up to the <math>m^{th}</math> random gate.
** The <math>(m+1)^{th}</math> gate is chosen to be the inverse of the composition of the first <math>m</math> random gates and interlaced <math>C</math> gates.
** The rest of the steps remain the same and finally after plotting the new average sequence fidelity with the sequence length and fitting it into either the gate and time dependent or the gate and time independent model, we receive the new depolarizing parameter obtained is <math>p_c</math>, which replaces <math>p</math>.
** The new gate error is calculated as <math>r_c = \frac{(d-1)(1-p_c/p)}{d}</math>
* Wallman, Granade, Harper, F., NJP 2015 [[Purity benchmarking]]: A unitarity can be estimated via purity benchmarking, which is an RB-like experiment that estimates a decay rate.

==Related Papers==
* E.Knill et al (2007) arXiv:0707.0963: gate and time-independent noise model
* E. Mageson et al (2011) arXiv:1009.3639: multi-parameter model
* Magesan et al. PRL (2012): Interleaved Randomized Benchmarking
* Harper et al (2016) arXiv:1608.02943v2: Interleaved Randomised Benchmarking to estimate fidelity of T gates
* Wallman, Granade, Harper, F., NJP 2015: Purity benchmarking

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Standard Randomised Benchmarking

2020-03-27T13:37:58Z

Rhea: /* Procedure Description */

[https://arxiv.org/abs/1109.6887v2 Randomized benchmarking] is a protocol that yields estimates of the computationally relevant errors without relying on accurate quantum state preparation and measurement. This is used to determine the error probability per gate in computational context and also gives an overall [[average fidelity]] for the noise in the gates.

'''Tags:''' [[:Category: Certification protocol|Certification Protocol]], [[:Category: Average gate fidelity|Average gate fidelity]], [[Randomised Benchmarking]], Clifford group

==Assumptions==
* The measurements performed are trusted.
* Noise model can be assumed to be gate and time-dependent or gate and time-independent.
* The noise model is independent and identically distributed (IID).

==Outline==
Randomized benchmarking method involves applying many random sequences of gates of varying lengths to a standard initial state. Each sequence ends with a randomized measurement that determines whether the correct final state was obtained. The average computationally relevant error per gate is obtained from the increase in error probability of the final measurements as a function of sequence length.

The random gates are taken from the [[Clifford group]]. The restriction to the Clifford group ensures that the measurements can be of one-qubit Pauli operators that yield at least one deterministic one-bit answer in the absence of errors.

This method consists of the following steps:

* A fixed sequence length is selected which is smaller than a predefined maximum sequence length. A random sequence of this length is chosen from the Clifford group.
* The operations are applied to the initial state corresponding to the selected sequence and then a final operator is applied which inverts all the previous operations.
* The final state is then measured to check if it matches the initial state. This process is performed several times with the same sequence to estimate the survival probability (the probability that the final state which returns to its initial state).
* Other random sequences of the same fixed sequence length are picked and the above-mentioned process is repeated to calculate the corresponding survival probability. This is used to calculate the average survival probability for the sequence length.
* The same procedure is repeated for multiple different sequence lengths.
* The observed survival probabilities are then plotted against the sequence length and then this is fit to an exponential decay curve, which is used to estimate the fidelity and also to calculate the average error rate which is the metric for randomized benchmarking.

==Hardware Requirements==
* Quantum computational resources to perform Clifford gates.
* Trusted Measurement device.

==Notation==
* <math>p</math>: Depolarizing parameter
* <math>d</math>: Dimension of Hilbert space
* <math>F_{avg}</math>: Average fidelity, <math>F_{avg} = p + \frac{1-p}{d}</math>
* <math>r</math>: Average error rate, <math>r = 1- F_{avg}, r = \frac{(d-1)(1-p)}{d}</math>
* <math>m</math>: Selected sequence length
* <math>K_m</math>: Total randomly selected sequence of <math>m</math> sequence length
* Clif<math>_n</math>: Clifford group
* C<math>_i</math>: Random element of Clifford group
* <math>S_{(i_1, ...,i_m)}</math> = <math>S_{\mathbf{i_m}}</math>: Random sequence of operations of length <math>m</math>
* <math>M</math>: Number of different data points to get the error model
* <math>\Lambda_{i,j}</math>: Implementation of C<math>_i</math> at time j (1 <math>\leq</math> j <math>\leq</math> M) results in this error map. <math>\Lambda_{i,1}, ..., \Lambda_{i,M}</math> are the different time-dependent noise operators affecting C<math>_i</math>.
* <math>|\psi\rangle</math>: initial state
* <math>E_{\psi}</math>: POVM element which takes into account the measurement error.
* <math>F_{seq}(m, \psi) = Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>: Survival probability of a sequence. <math>\rho_\psi</math> is a quantum state that takes into account errors in preparing <math>\langle \psi |\psi \rangle</math>
* <math>F_g^{(0)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time independent error model
* <math>F_g^{(1)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time dependent error model. In this model, the parameter <math>(q-p^2)</math> is a measure of the degree of gate-dependence in the error.
* <math>A_0, B_0</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time independent error model
* <math>A_1, B_1, C_1</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time dependent error model.
* <math>R_{m+1}</math>: <math>\frac{1}{|Clif_n|}\sum_i\Lambda_{i, m+1} \otimes (C_i \otimes \Lambda \otimes C_i^{\dagger})</math>

==Properties==
* '''Figure of merit''': average error rate, average gate fidelity
* The errors which are considered here are State preparation and measurement errors, error on the final gate, which are gate and time-independent errors. Gate and time-dependent errors can also be taken into consideration. This method is insensitive to SPAM error.
* The random gates are picked from the Clifford group.
* For noise estimation, the uniform probability distribution over Clifford group comprises a [[unitary 2-design]].
* This protocol provides a scalable method for benchmarking the set of Clifford gates.
* To obtain a more accurate value for <math>p</math> one should always use the first order fitting model unless prior knowledge of the noise indicates that it is effectively gate-independent.

==Procedure Description==
'''Output''': Figure of merit: <math>r</math>

* For <math>1, 2, ..., M</math>:
** Pick random sequence length <math>m</math>
** For <math>k = 1, 2, ..., K_m</math> sequences:
*** For <math>j = 1, 2 ..., m+1</math>:
**** If <math>j == m+1</math>, apply inverse operator of previous operations
**** else, apply random operation C<math>_i</math>
*** Thus, <math>S_{\mathbf{i_m}} = \bigotimes^{m+1}_{j=1} (\Lambda_{(i_j, j)} C_{i_j})</math> and <math>i_{m+1}</math> is uniquely determined by <math>(i_1, ...,i_m)</math>
*** Measure survival probability <math>Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>
** Estimate average survival probability <math>Tr[E_{\psi}S_{\mathbf{K_m}}(\rho_\psi)]</math> over all <math>K_m</math> sequences, where <math>S_{\mathbf{K_m}} = \frac{1}{K_m}\sum_{i_m} S_{i_m}</math>
* Fit the results for the averaged sequence fidelity for all <math>m</math> into the models:
** For gate and time independent error model:
*** <math>F_g^{(0)}(m, |\psi\rangle) = A_0p^m + B_0</math>
** For gate and time dependent error model:
*** <math>F_g^{(1)}(m, |\psi\rangle) = A_1p^m + B_1 + C_1(m-1)(q-p^2)p^{m-2}</math>
* <math>p</math> is extracted from the model and <math>r</math> is estimated, <math>r = \frac{(d-1)(1-p)}{d}</math>

==Further Information==
* Fitting models are described and derived as seen in [https://arxiv.org/abs/1109.6887v2 E. Mageson et al]. The coefficients derived are:
** <math>A_0</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
** <math>B_0</math> = Tr<math>[E_\psi \Lambda(\frac{\mathbb{1}}{d})]</math>
** <math>A_1</math> = Tr<math>[E_\psi \Lambda(\frac{Q_1\rho_\psi}{p} - \rho_\psi + \frac{(p-1)\mathbb{1}}{pd})]</math> + Tr<math>[E_\psi R_{m+1}(\frac{\rho_\psi}{p} - \frac{\mathbb{1}}{pd})]</math>
** <math>B_1</math> = Tr<math>[E_\psi R_{m+1}(\frac{\mathbb{1}}{d})]</math>
** <math>C_1</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
* The case where Randomized benchmarking fails: Suppose the noise is time dependent and for each <math>i, \Lambda_i = C_i^{\dagger}</math>. Then <math>F_g(m, \psi) = 1</math> for every <math>m</math> even though there is a substantial error on each <math>C_i</math> and so benchmarking fails.
* [https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.109.080505 Interleaved Randomized Benchmarking]: This protocol consists of interleaving random Clifford gates between the gate of interest and provides an estimate as well as theoretical bounds for the average error of the gate under test, so long as the average noise variation over all Clifford gates is small. Here the procedure followed is:
** Choose <math>K</math> sequences of Clifford elements where the first Clifford <math>C_{i_1}</math> in each sequence is chosen uniformly at random from Clif<math>_n</math>, the second is always chosen to be <math>C</math>(gate of interest), and alternate between uniformly random Clifford elements and deterministic <math>C</math> up to the <math>m^{th}</math> random gate.
** The <math>(m+1)^{th}</math> gate is chosen to be the inverse of the composition of the first <math>m</math> random gates and interlaced <math>C</math> gates.
** The rest of the steps remain the same and finally after plotting the new average sequence fidelity with the sequence length and fitting it into either the gate and time dependent or the gate and time independent model, we receive the new depolarizing parameter obtained is <math>p_c</math>, which replaces <math>p</math>.
** The new gate error is calculated as <math>r_c = \frac{(d-1)(1-p_c/p)}{d}</math>
* Wallman, Granade, Harper, F., NJP 2015 [[Purity benchmarking]]: A unitarity can be estimated via purity benchmarking, which is an RB-like experiment that estimates a decay rate.

==Related Papers==
* E.Knill et al (2007) arXiv:0707.0963: gate and time-independent noise model
* E. Mageson et al (2011) arXiv:1009.3639: multi-parameter model
* Magesan et al. PRL (2012): Interleaved Randomized Benchmarking
* Harper et al (2016) arXiv:1608.02943v2: Interleaved Randomised Benchmarking to estimate fidelity of T gates
* Wallman, Granade, Harper, F., NJP 2015: Purity benchmarking

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Standard Randomised Benchmarking

2020-03-27T13:37:44Z

Rhea: /* Procedure Description */

[https://arxiv.org/abs/1109.6887v2 Randomized benchmarking] is a protocol that yields estimates of the computationally relevant errors without relying on accurate quantum state preparation and measurement. This is used to determine the error probability per gate in computational context and also gives an overall [[average fidelity]] for the noise in the gates.

'''Tags:''' [[:Category: Certification protocol|Certification Protocol]], [[:Category: Average gate fidelity|Average gate fidelity]], [[Randomised Benchmarking]], Clifford group

==Assumptions==
* The measurements performed are trusted.
* Noise model can be assumed to be gate and time-dependent or gate and time-independent.
* The noise model is independent and identically distributed (IID).

==Outline==
Randomized benchmarking method involves applying many random sequences of gates of varying lengths to a standard initial state. Each sequence ends with a randomized measurement that determines whether the correct final state was obtained. The average computationally relevant error per gate is obtained from the increase in error probability of the final measurements as a function of sequence length.

The random gates are taken from the [[Clifford group]]. The restriction to the Clifford group ensures that the measurements can be of one-qubit Pauli operators that yield at least one deterministic one-bit answer in the absence of errors.

This method consists of the following steps:

* A fixed sequence length is selected which is smaller than a predefined maximum sequence length. A random sequence of this length is chosen from the Clifford group.
* The operations are applied to the initial state corresponding to the selected sequence and then a final operator is applied which inverts all the previous operations.
* The final state is then measured to check if it matches the initial state. This process is performed several times with the same sequence to estimate the survival probability (the probability that the final state which returns to its initial state).
* Other random sequences of the same fixed sequence length are picked and the above-mentioned process is repeated to calculate the corresponding survival probability. This is used to calculate the average survival probability for the sequence length.
* The same procedure is repeated for multiple different sequence lengths.
* The observed survival probabilities are then plotted against the sequence length and then this is fit to an exponential decay curve, which is used to estimate the fidelity and also to calculate the average error rate which is the metric for randomized benchmarking.

==Hardware Requirements==
* Quantum computational resources to perform Clifford gates.
* Trusted Measurement device.

==Notation==
* <math>p</math>: Depolarizing parameter
* <math>d</math>: Dimension of Hilbert space
* <math>F_{avg}</math>: Average fidelity, <math>F_{avg} = p + \frac{1-p}{d}</math>
* <math>r</math>: Average error rate, <math>r = 1- F_{avg}, r = \frac{(d-1)(1-p)}{d}</math>
* <math>m</math>: Selected sequence length
* <math>K_m</math>: Total randomly selected sequence of <math>m</math> sequence length
* Clif<math>_n</math>: Clifford group
* C<math>_i</math>: Random element of Clifford group
* <math>S_{(i_1, ...,i_m)}</math> = <math>S_{\mathbf{i_m}}</math>: Random sequence of operations of length <math>m</math>
* <math>M</math>: Number of different data points to get the error model
* <math>\Lambda_{i,j}</math>: Implementation of C<math>_i</math> at time j (1 <math>\leq</math> j <math>\leq</math> M) results in this error map. <math>\Lambda_{i,1}, ..., \Lambda_{i,M}</math> are the different time-dependent noise operators affecting C<math>_i</math>.
* <math>|\psi\rangle</math>: initial state
* <math>E_{\psi}</math>: POVM element which takes into account the measurement error.
* <math>F_{seq}(m, \psi) = Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>: Survival probability of a sequence. <math>\rho_\psi</math> is a quantum state that takes into account errors in preparing <math>\langle \psi |\psi \rangle</math>
* <math>F_g^{(0)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time independent error model
* <math>F_g^{(1)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time dependent error model. In this model, the parameter <math>(q-p^2)</math> is a measure of the degree of gate-dependence in the error.
* <math>A_0, B_0</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time independent error model
* <math>A_1, B_1, C_1</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time dependent error model.
* <math>R_{m+1}</math>: <math>\frac{1}{|Clif_n|}\sum_i\Lambda_{i, m+1} \otimes (C_i \otimes \Lambda \otimes C_i^{\dagger})</math>

==Properties==
* '''Figure of merit''': average error rate, average gate fidelity
* The errors which are considered here are State preparation and measurement errors, error on the final gate, which are gate and time-independent errors. Gate and time-dependent errors can also be taken into consideration. This method is insensitive to SPAM error.
* The random gates are picked from the Clifford group.
* For noise estimation, the uniform probability distribution over Clifford group comprises a [[unitary 2-design]].
* This protocol provides a scalable method for benchmarking the set of Clifford gates.
* To obtain a more accurate value for <math>p</math> one should always use the first order fitting model unless prior knowledge of the noise indicates that it is effectively gate-independent.

==Procedure Description==
'''Output''': Figure of merit: <math>r</math>

* For <math>1, 2, ..., M-1</math>:
** Pick random sequence length <math>m</math>
** For <math>k = 1, 2, ..., K_m</math> sequences:
*** For <math>j = 1, 2 ..., m+1</math>:
**** If <math>j == m+1</math>, apply inverse operator of previous operations
**** else, apply random operation C<math>_i</math>
*** Thus, <math>S_{\mathbf{i_m}} = \bigotimes^{m+1}_{j=1} (\Lambda_{(i_j, j)} C_{i_j})</math> and <math>i_{m+1}</math> is uniquely determined by <math>(i_1, ...,i_m)</math>
*** Measure survival probability <math>Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>
** Estimate average survival probability <math>Tr[E_{\psi}S_{\mathbf{K_m}}(\rho_\psi)]</math> over all <math>K_m</math> sequences, where <math>S_{\mathbf{K_m}} = \frac{1}{K_m}\sum_{i_m} S_{i_m}</math>
* Fit the results for the averaged sequence fidelity for all <math>m</math> into the models:
** For gate and time independent error model:
*** <math>F_g^{(0)}(m, |\psi\rangle) = A_0p^m + B_0</math>
** For gate and time dependent error model:
*** <math>F_g^{(1)}(m, |\psi\rangle) = A_1p^m + B_1 + C_1(m-1)(q-p^2)p^{m-2}</math>
* <math>p</math> is extracted from the model and <math>r</math> is estimated, <math>r = \frac{(d-1)(1-p)}{d}</math>

==Further Information==
* Fitting models are described and derived as seen in [https://arxiv.org/abs/1109.6887v2 E. Mageson et al]. The coefficients derived are:
** <math>A_0</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
** <math>B_0</math> = Tr<math>[E_\psi \Lambda(\frac{\mathbb{1}}{d})]</math>
** <math>A_1</math> = Tr<math>[E_\psi \Lambda(\frac{Q_1\rho_\psi}{p} - \rho_\psi + \frac{(p-1)\mathbb{1}}{pd})]</math> + Tr<math>[E_\psi R_{m+1}(\frac{\rho_\psi}{p} - \frac{\mathbb{1}}{pd})]</math>
** <math>B_1</math> = Tr<math>[E_\psi R_{m+1}(\frac{\mathbb{1}}{d})]</math>
** <math>C_1</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
* The case where Randomized benchmarking fails: Suppose the noise is time dependent and for each <math>i, \Lambda_i = C_i^{\dagger}</math>. Then <math>F_g(m, \psi) = 1</math> for every <math>m</math> even though there is a substantial error on each <math>C_i</math> and so benchmarking fails.
* [https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.109.080505 Interleaved Randomized Benchmarking]: This protocol consists of interleaving random Clifford gates between the gate of interest and provides an estimate as well as theoretical bounds for the average error of the gate under test, so long as the average noise variation over all Clifford gates is small. Here the procedure followed is:
** Choose <math>K</math> sequences of Clifford elements where the first Clifford <math>C_{i_1}</math> in each sequence is chosen uniformly at random from Clif<math>_n</math>, the second is always chosen to be <math>C</math>(gate of interest), and alternate between uniformly random Clifford elements and deterministic <math>C</math> up to the <math>m^{th}</math> random gate.
** The <math>(m+1)^{th}</math> gate is chosen to be the inverse of the composition of the first <math>m</math> random gates and interlaced <math>C</math> gates.
** The rest of the steps remain the same and finally after plotting the new average sequence fidelity with the sequence length and fitting it into either the gate and time dependent or the gate and time independent model, we receive the new depolarizing parameter obtained is <math>p_c</math>, which replaces <math>p</math>.
** The new gate error is calculated as <math>r_c = \frac{(d-1)(1-p_c/p)}{d}</math>
* Wallman, Granade, Harper, F., NJP 2015 [[Purity benchmarking]]: A unitarity can be estimated via purity benchmarking, which is an RB-like experiment that estimates a decay rate.

==Related Papers==
* E.Knill et al (2007) arXiv:0707.0963: gate and time-independent noise model
* E. Mageson et al (2011) arXiv:1009.3639: multi-parameter model
* Magesan et al. PRL (2012): Interleaved Randomized Benchmarking
* Harper et al (2016) arXiv:1608.02943v2: Interleaved Randomised Benchmarking to estimate fidelity of T gates
* Wallman, Granade, Harper, F., NJP 2015: Purity benchmarking

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Standard Randomised Benchmarking

2020-03-27T13:37:08Z

Rhea: /* Notation */

[https://arxiv.org/abs/1109.6887v2 Randomized benchmarking] is a protocol that yields estimates of the computationally relevant errors without relying on accurate quantum state preparation and measurement. This is used to determine the error probability per gate in computational context and also gives an overall [[average fidelity]] for the noise in the gates.

'''Tags:''' [[:Category: Certification protocol|Certification Protocol]], [[:Category: Average gate fidelity|Average gate fidelity]], [[Randomised Benchmarking]], Clifford group

==Assumptions==
* The measurements performed are trusted.
* Noise model can be assumed to be gate and time-dependent or gate and time-independent.
* The noise model is independent and identically distributed (IID).

==Outline==
Randomized benchmarking method involves applying many random sequences of gates of varying lengths to a standard initial state. Each sequence ends with a randomized measurement that determines whether the correct final state was obtained. The average computationally relevant error per gate is obtained from the increase in error probability of the final measurements as a function of sequence length.

The random gates are taken from the [[Clifford group]]. The restriction to the Clifford group ensures that the measurements can be of one-qubit Pauli operators that yield at least one deterministic one-bit answer in the absence of errors.

This method consists of the following steps:

* A fixed sequence length is selected which is smaller than a predefined maximum sequence length. A random sequence of this length is chosen from the Clifford group.
* The operations are applied to the initial state corresponding to the selected sequence and then a final operator is applied which inverts all the previous operations.
* The final state is then measured to check if it matches the initial state. This process is performed several times with the same sequence to estimate the survival probability (the probability that the final state which returns to its initial state).
* Other random sequences of the same fixed sequence length are picked and the above-mentioned process is repeated to calculate the corresponding survival probability. This is used to calculate the average survival probability for the sequence length.
* The same procedure is repeated for multiple different sequence lengths.
* The observed survival probabilities are then plotted against the sequence length and then this is fit to an exponential decay curve, which is used to estimate the fidelity and also to calculate the average error rate which is the metric for randomized benchmarking.

==Hardware Requirements==
* Quantum computational resources to perform Clifford gates.
* Trusted Measurement device.

==Notation==
* <math>p</math>: Depolarizing parameter
* <math>d</math>: Dimension of Hilbert space
* <math>F_{avg}</math>: Average fidelity, <math>F_{avg} = p + \frac{1-p}{d}</math>
* <math>r</math>: Average error rate, <math>r = 1- F_{avg}, r = \frac{(d-1)(1-p)}{d}</math>
* <math>m</math>: Selected sequence length
* <math>K_m</math>: Total randomly selected sequence of <math>m</math> sequence length
* Clif<math>_n</math>: Clifford group
* C<math>_i</math>: Random element of Clifford group
* <math>S_{(i_1, ...,i_m)}</math> = <math>S_{\mathbf{i_m}}</math>: Random sequence of operations of length <math>m</math>
* <math>M</math>: Number of different data points to get the error model
* <math>\Lambda_{i,j}</math>: Implementation of C<math>_i</math> at time j (1 <math>\leq</math> j <math>\leq</math> M) results in this error map. <math>\Lambda_{i,1}, ..., \Lambda_{i,M}</math> are the different time-dependent noise operators affecting C<math>_i</math>.
* <math>|\psi\rangle</math>: initial state
* <math>E_{\psi}</math>: POVM element which takes into account the measurement error.
* <math>F_{seq}(m, \psi) = Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>: Survival probability of a sequence. <math>\rho_\psi</math> is a quantum state that takes into account errors in preparing <math>\langle \psi |\psi \rangle</math>
* <math>F_g^{(0)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time independent error model
* <math>F_g^{(1)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time dependent error model. In this model, the parameter <math>(q-p^2)</math> is a measure of the degree of gate-dependence in the error.
* <math>A_0, B_0</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time independent error model
* <math>A_1, B_1, C_1</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time dependent error model.
* <math>R_{m+1}</math>: <math>\frac{1}{|Clif_n|}\sum_i\Lambda_{i, m+1} \otimes (C_i \otimes \Lambda \otimes C_i^{\dagger})</math>

==Properties==
* '''Figure of merit''': average error rate, average gate fidelity
* The errors which are considered here are State preparation and measurement errors, error on the final gate, which are gate and time-independent errors. Gate and time-dependent errors can also be taken into consideration. This method is insensitive to SPAM error.
* The random gates are picked from the Clifford group.
* For noise estimation, the uniform probability distribution over Clifford group comprises a [[unitary 2-design]].
* This protocol provides a scalable method for benchmarking the set of Clifford gates.
* To obtain a more accurate value for <math>p</math> one should always use the first order fitting model unless prior knowledge of the noise indicates that it is effectively gate-independent.

==Procedure Description==
'''Output''': Figure of merit: <math>r</math>

* For <math>m = 1, 2, ..., M-1</math>:
** For <math>k = 1, 2, ..., K_m</math> sequences:
*** For <math>j = 1, 2 ..., m+1</math>:
**** If <math>j == m+1</math>, apply inverse operator of previous operations
**** else, apply random operation C<math>_i</math>
*** Thus, <math>S_{\mathbf{i_m}} = \bigotimes^{m+1}_{j=1} (\Lambda_{(i_j, j)} C_{i_j})</math> and <math>i_{m+1}</math> is uniquely determined by <math>(i_1, ...,i_m)</math>
*** Measure survival probability <math>Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>
** Estimate average survival probability <math>Tr[E_{\psi}S_{\mathbf{K_m}}(\rho_\psi)]</math> over all <math>K_m</math> sequences, where <math>S_{\mathbf{K_m}} = \frac{1}{K_m}\sum_{i_m} S_{i_m}</math>
* Fit the results for the averaged sequence fidelity for all <math>m</math> into the models:
** For gate and time independent error model:
*** <math>F_g^{(0)}(m, |\psi\rangle) = A_0p^m + B_0</math>
** For gate and time dependent error model:
*** <math>F_g^{(1)}(m, |\psi\rangle) = A_1p^m + B_1 + C_1(m-1)(q-p^2)p^{m-2}</math>
* <math>p</math> is extracted from the model and <math>r</math> is estimated, <math>r = \frac{(d-1)(1-p)}{d}</math>

==Further Information==
* Fitting models are described and derived as seen in [https://arxiv.org/abs/1109.6887v2 E. Mageson et al]. The coefficients derived are:
** <math>A_0</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
** <math>B_0</math> = Tr<math>[E_\psi \Lambda(\frac{\mathbb{1}}{d})]</math>
** <math>A_1</math> = Tr<math>[E_\psi \Lambda(\frac{Q_1\rho_\psi}{p} - \rho_\psi + \frac{(p-1)\mathbb{1}}{pd})]</math> + Tr<math>[E_\psi R_{m+1}(\frac{\rho_\psi}{p} - \frac{\mathbb{1}}{pd})]</math>
** <math>B_1</math> = Tr<math>[E_\psi R_{m+1}(\frac{\mathbb{1}}{d})]</math>
** <math>C_1</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
* The case where Randomized benchmarking fails: Suppose the noise is time dependent and for each <math>i, \Lambda_i = C_i^{\dagger}</math>. Then <math>F_g(m, \psi) = 1</math> for every <math>m</math> even though there is a substantial error on each <math>C_i</math> and so benchmarking fails.
* [https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.109.080505 Interleaved Randomized Benchmarking]: This protocol consists of interleaving random Clifford gates between the gate of interest and provides an estimate as well as theoretical bounds for the average error of the gate under test, so long as the average noise variation over all Clifford gates is small. Here the procedure followed is:
** Choose <math>K</math> sequences of Clifford elements where the first Clifford <math>C_{i_1}</math> in each sequence is chosen uniformly at random from Clif<math>_n</math>, the second is always chosen to be <math>C</math>(gate of interest), and alternate between uniformly random Clifford elements and deterministic <math>C</math> up to the <math>m^{th}</math> random gate.
** The <math>(m+1)^{th}</math> gate is chosen to be the inverse of the composition of the first <math>m</math> random gates and interlaced <math>C</math> gates.
** The rest of the steps remain the same and finally after plotting the new average sequence fidelity with the sequence length and fitting it into either the gate and time dependent or the gate and time independent model, we receive the new depolarizing parameter obtained is <math>p_c</math>, which replaces <math>p</math>.
** The new gate error is calculated as <math>r_c = \frac{(d-1)(1-p_c/p)}{d}</math>
* Wallman, Granade, Harper, F., NJP 2015 [[Purity benchmarking]]: A unitarity can be estimated via purity benchmarking, which is an RB-like experiment that estimates a decay rate.

==Related Papers==
* E.Knill et al (2007) arXiv:0707.0963: gate and time-independent noise model
* E. Mageson et al (2011) arXiv:1009.3639: multi-parameter model
* Magesan et al. PRL (2012): Interleaved Randomized Benchmarking
* Harper et al (2016) arXiv:1608.02943v2: Interleaved Randomised Benchmarking to estimate fidelity of T gates
* Wallman, Granade, Harper, F., NJP 2015: Purity benchmarking

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Standard Randomised Benchmarking

2020-03-27T13:33:24Z

Rhea: /* Procedure Description */

[https://arxiv.org/abs/1109.6887v2 Randomized benchmarking] is a protocol that yields estimates of the computationally relevant errors without relying on accurate quantum state preparation and measurement. This is used to determine the error probability per gate in computational context and also gives an overall [[average fidelity]] for the noise in the gates.

'''Tags:''' [[:Category: Certification protocol|Certification Protocol]], [[:Category: Average gate fidelity|Average gate fidelity]], [[Randomised Benchmarking]], Clifford group

==Assumptions==
* The measurements performed are trusted.
* Noise model can be assumed to be gate and time-dependent or gate and time-independent.
* The noise model is independent and identically distributed (IID).

==Outline==
Randomized benchmarking method involves applying many random sequences of gates of varying lengths to a standard initial state. Each sequence ends with a randomized measurement that determines whether the correct final state was obtained. The average computationally relevant error per gate is obtained from the increase in error probability of the final measurements as a function of sequence length.

The random gates are taken from the [[Clifford group]]. The restriction to the Clifford group ensures that the measurements can be of one-qubit Pauli operators that yield at least one deterministic one-bit answer in the absence of errors.

This method consists of the following steps:

* A fixed sequence length is selected which is smaller than a predefined maximum sequence length. A random sequence of this length is chosen from the Clifford group.
* The operations are applied to the initial state corresponding to the selected sequence and then a final operator is applied which inverts all the previous operations.
* The final state is then measured to check if it matches the initial state. This process is performed several times with the same sequence to estimate the survival probability (the probability that the final state which returns to its initial state).
* Other random sequences of the same fixed sequence length are picked and the above-mentioned process is repeated to calculate the corresponding survival probability. This is used to calculate the average survival probability for the sequence length.
* The same procedure is repeated for multiple different sequence lengths.
* The observed survival probabilities are then plotted against the sequence length and then this is fit to an exponential decay curve, which is used to estimate the fidelity and also to calculate the average error rate which is the metric for randomized benchmarking.

==Hardware Requirements==
* Quantum computational resources to perform Clifford gates.
* Trusted Measurement device.

==Notation==
* <math>p</math>: Depolarizing parameter
* <math>d</math>: Dimension of Hilbert space
* <math>F_{avg}</math>: Average fidelity, <math>F_{avg} = p + \frac{1-p}{d}</math>
* <math>r</math>: Average error rate, <math>r = 1- F_{avg}, r = \frac{(d-1)(1-p)}{d}</math>
* <math>m</math>: Selected sequence length
* <math>K_m</math>: Total randomly selected sequence of <math>m</math> sequence length
* Clif<math>_n</math>: Clifford group
* C<math>_i</math>: Random element of Clifford group
* <math>S_{(i_1, ...,i_m)}</math> = <math>S_{\mathbf{i_m}}</math>: Random sequence of operations of length <math>m</math>
* <math>M</math>: Maximum sequence length of applying Clifford group Clif<math>_n</math>
* <math>\Lambda_{i,j}</math>: Implementation of C<math>_i</math> at time j (1 <math>\leq</math> j <math>\leq</math> M) results in this error map. <math>\Lambda_{i,1}, ..., \Lambda_{i,M}</math> are the different time-dependent noise operators affecting C<math>_i</math>.
* <math>|\psi\rangle</math>: initial state
* <math>E_{\psi}</math>: POVM element which takes into account the measurement error.
* <math>F_{seq}(m, \psi) = Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>: Survival probability of a sequence. <math>\rho_\psi</math> is a quantum state that takes into account errors in preparing <math>\langle \psi |\psi \rangle</math>
* <math>F_g^{(0)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time independent error model
* <math>F_g^{(1)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time dependent error model. In this model, the parameter <math>(q-p^2)</math> is a measure of the degree of gate-dependence in the error.
* <math>A_0, B_0</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time independent error model
* <math>A_1, B_1, C_1</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time dependent error model.
* <math>R_{m+1}</math>: <math>\frac{1}{|Clif_n|}\sum_i\Lambda_{i, m+1} \otimes (C_i \otimes \Lambda \otimes C_i^{\dagger})</math>

==Properties==
* '''Figure of merit''': average error rate, average gate fidelity
* The errors which are considered here are State preparation and measurement errors, error on the final gate, which are gate and time-independent errors. Gate and time-dependent errors can also be taken into consideration. This method is insensitive to SPAM error.
* The random gates are picked from the Clifford group.
* For noise estimation, the uniform probability distribution over Clifford group comprises a [[unitary 2-design]].
* This protocol provides a scalable method for benchmarking the set of Clifford gates.
* To obtain a more accurate value for <math>p</math> one should always use the first order fitting model unless prior knowledge of the noise indicates that it is effectively gate-independent.

==Procedure Description==
'''Output''': Figure of merit: <math>r</math>

* For <math>m = 1, 2, ..., M-1</math>:
** For <math>k = 1, 2, ..., K_m</math> sequences:
*** For <math>j = 1, 2 ..., m+1</math>:
**** If <math>j == m+1</math>, apply inverse operator of previous operations
**** else, apply random operation C<math>_i</math>
*** Thus, <math>S_{\mathbf{i_m}} = \bigotimes^{m+1}_{j=1} (\Lambda_{(i_j, j)} C_{i_j})</math> and <math>i_{m+1}</math> is uniquely determined by <math>(i_1, ...,i_m)</math>
*** Measure survival probability <math>Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>
** Estimate average survival probability <math>Tr[E_{\psi}S_{\mathbf{K_m}}(\rho_\psi)]</math> over all <math>K_m</math> sequences, where <math>S_{\mathbf{K_m}} = \frac{1}{K_m}\sum_{i_m} S_{i_m}</math>
* Fit the results for the averaged sequence fidelity for all <math>m</math> into the models:
** For gate and time independent error model:
*** <math>F_g^{(0)}(m, |\psi\rangle) = A_0p^m + B_0</math>
** For gate and time dependent error model:
*** <math>F_g^{(1)}(m, |\psi\rangle) = A_1p^m + B_1 + C_1(m-1)(q-p^2)p^{m-2}</math>
* <math>p</math> is extracted from the model and <math>r</math> is estimated, <math>r = \frac{(d-1)(1-p)}{d}</math>

==Further Information==
* Fitting models are described and derived as seen in [https://arxiv.org/abs/1109.6887v2 E. Mageson et al]. The coefficients derived are:
** <math>A_0</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
** <math>B_0</math> = Tr<math>[E_\psi \Lambda(\frac{\mathbb{1}}{d})]</math>
** <math>A_1</math> = Tr<math>[E_\psi \Lambda(\frac{Q_1\rho_\psi}{p} - \rho_\psi + \frac{(p-1)\mathbb{1}}{pd})]</math> + Tr<math>[E_\psi R_{m+1}(\frac{\rho_\psi}{p} - \frac{\mathbb{1}}{pd})]</math>
** <math>B_1</math> = Tr<math>[E_\psi R_{m+1}(\frac{\mathbb{1}}{d})]</math>
** <math>C_1</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
* The case where Randomized benchmarking fails: Suppose the noise is time dependent and for each <math>i, \Lambda_i = C_i^{\dagger}</math>. Then <math>F_g(m, \psi) = 1</math> for every <math>m</math> even though there is a substantial error on each <math>C_i</math> and so benchmarking fails.
* [https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.109.080505 Interleaved Randomized Benchmarking]: This protocol consists of interleaving random Clifford gates between the gate of interest and provides an estimate as well as theoretical bounds for the average error of the gate under test, so long as the average noise variation over all Clifford gates is small. Here the procedure followed is:
** Choose <math>K</math> sequences of Clifford elements where the first Clifford <math>C_{i_1}</math> in each sequence is chosen uniformly at random from Clif<math>_n</math>, the second is always chosen to be <math>C</math>(gate of interest), and alternate between uniformly random Clifford elements and deterministic <math>C</math> up to the <math>m^{th}</math> random gate.
** The <math>(m+1)^{th}</math> gate is chosen to be the inverse of the composition of the first <math>m</math> random gates and interlaced <math>C</math> gates.
** The rest of the steps remain the same and finally after plotting the new average sequence fidelity with the sequence length and fitting it into either the gate and time dependent or the gate and time independent model, we receive the new depolarizing parameter obtained is <math>p_c</math>, which replaces <math>p</math>.
** The new gate error is calculated as <math>r_c = \frac{(d-1)(1-p_c/p)}{d}</math>
* Wallman, Granade, Harper, F., NJP 2015 [[Purity benchmarking]]: A unitarity can be estimated via purity benchmarking, which is an RB-like experiment that estimates a decay rate.

==Related Papers==
* E.Knill et al (2007) arXiv:0707.0963: gate and time-independent noise model
* E. Mageson et al (2011) arXiv:1009.3639: multi-parameter model
* Magesan et al. PRL (2012): Interleaved Randomized Benchmarking
* Harper et al (2016) arXiv:1608.02943v2: Interleaved Randomised Benchmarking to estimate fidelity of T gates
* Wallman, Granade, Harper, F., NJP 2015: Purity benchmarking

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Standard Randomised Benchmarking

2020-03-24T10:52:49Z

Rhea: /* Properties */

[https://arxiv.org/abs/1109.6887v2 Randomized benchmarking] is a protocol that yields estimates of the computationally relevant errors without relying on accurate quantum state preparation and measurement. This is used to determine the error probability per gate in computational context and also gives an overall [[average fidelity]] for the noise in the gates.

'''Tags:''' [[:Category: Certification protocol|Certification Protocol]], [[:Category: Average gate fidelity|Average gate fidelity]], [[Randomised Benchmarking]], Clifford group

==Assumptions==
* The measurements performed are trusted.
* Noise model can be assumed to be gate and time-dependent or gate and time-independent.
* The noise model is independent and identically distributed (IID).

==Outline==
Randomized benchmarking method involves applying many random sequences of gates of varying lengths to a standard initial state. Each sequence ends with a randomized measurement that determines whether the correct final state was obtained. The average computationally relevant error per gate is obtained from the increase in error probability of the final measurements as a function of sequence length.

The random gates are taken from the [[Clifford group]]. The restriction to the Clifford group ensures that the measurements can be of one-qubit Pauli operators that yield at least one deterministic one-bit answer in the absence of errors.

This method consists of the following steps:

* A fixed sequence length is selected which is smaller than a predefined maximum sequence length. A random sequence of this length is chosen from the Clifford group.
* The operations are applied to the initial state corresponding to the selected sequence and then a final operator is applied which inverts all the previous operations.
* The final state is then measured to check if it matches the initial state. This process is performed several times with the same sequence to estimate the survival probability (the probability that the final state which returns to its initial state).
* Other random sequences of the same fixed sequence length are picked and the above-mentioned process is repeated to calculate the corresponding survival probability. This is used to calculate the average survival probability for the sequence length.
* The same procedure is repeated for multiple different sequence lengths.
* The observed survival probabilities are then plotted against the sequence length and then this is fit to an exponential decay curve, which is used to estimate the fidelity and also to calculate the average error rate which is the metric for randomized benchmarking.

==Hardware Requirements==
* Quantum computational resources to perform Clifford gates.
* Trusted Measurement device.

==Notation==
* <math>p</math>: Depolarizing parameter
* <math>d</math>: Dimension of Hilbert space
* <math>F_{avg}</math>: Average fidelity, <math>F_{avg} = p + \frac{1-p}{d}</math>
* <math>r</math>: Average error rate, <math>r = 1- F_{avg}, r = \frac{(d-1)(1-p)}{d}</math>
* <math>m</math>: Selected sequence length
* <math>K_m</math>: Total randomly selected sequence of <math>m</math> sequence length
* Clif<math>_n</math>: Clifford group
* C<math>_i</math>: Random element of Clifford group
* <math>S_{(i_1, ...,i_m)}</math> = <math>S_{\mathbf{i_m}}</math>: Random sequence of operations of length <math>m</math>
* <math>M</math>: Maximum sequence length of applying Clifford group Clif<math>_n</math>
* <math>\Lambda_{i,j}</math>: Implementation of C<math>_i</math> at time j (1 <math>\leq</math> j <math>\leq</math> M) results in this error map. <math>\Lambda_{i,1}, ..., \Lambda_{i,M}</math> are the different time-dependent noise operators affecting C<math>_i</math>.
* <math>|\psi\rangle</math>: initial state
* <math>E_{\psi}</math>: POVM element which takes into account the measurement error.
* <math>F_{seq}(m, \psi) = Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>: Survival probability of a sequence. <math>\rho_\psi</math> is a quantum state that takes into account errors in preparing <math>\langle \psi |\psi \rangle</math>
* <math>F_g^{(0)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time independent error model
* <math>F_g^{(1)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time dependent error model. In this model, the parameter <math>(q-p^2)</math> is a measure of the degree of gate-dependence in the error.
* <math>A_0, B_0</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time independent error model
* <math>A_1, B_1, C_1</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time dependent error model.
* <math>R_{m+1}</math>: <math>\frac{1}{|Clif_n|}\sum_i\Lambda_{i, m+1} \otimes (C_i \otimes \Lambda \otimes C_i^{\dagger})</math>

==Properties==
* '''Figure of merit''': average error rate, average gate fidelity
* The errors which are considered here are State preparation and measurement errors, error on the final gate, which are gate and time-independent errors. Gate and time-dependent errors can also be taken into consideration. This method is insensitive to SPAM error.
* The random gates are picked from the Clifford group.
* For noise estimation, the uniform probability distribution over Clifford group comprises a [[unitary 2-design]].
* This protocol provides a scalable method for benchmarking the set of Clifford gates.
* To obtain a more accurate value for <math>p</math> one should always use the first order fitting model unless prior knowledge of the noise indicates that it is effectively gate-independent.

==Procedure Description==
'''Output''': Figure of merit: <math>r</math>

* For <math>m = 1, 2, ..., M-1</math>:
** For <math>k = 1, 2, ..., K_m</math> sequences:
*** For <math>j = 1, 2 ..., m+1</math>:
**** If <math>j == m+1</math>, apply inverse operator of previous operations
**** else, apply random operation C<math>_i</math>
*** Thus, <math>S_{\mathbf{i_m}} = \bigotimes^{m+1}_{j+1} (\Lambda_{(i_j, j)} C_i)</math> and <math>i_{m+1}</math> is uniquely determined by <math>(i_1, ...,i_m)</math>
*** Measure survival probability <math>Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>
** Estimate average survival probability <math>Tr[E_{\psi}S_{\mathbf{K_m}}(\rho_\psi)]</math> over all <math>K_m</math> sequences, where <math>S_{\mathbf{K_m}} = \frac{1}{K_m}\sum_{i_m} S_{i_m}</math>
* Fit the results for the averaged sequence fidelity for all <math>m</math> into the models:
** For gate and time independent error model:
*** <math>F_g^{(0)}(m, |\psi\rangle) = A_0p^m + B_0</math>
** For gate and time dependent error model:
*** <math>F_g^{(1)}(m, |\psi\rangle) = A_1p^m + B_1 + C_1(m-1)(q-p^2)p^{m-2}</math>
* <math>p</math> is extracted from the model and <math>r</math> is estimated, <math>r = \frac{(d-1)(1-p)}{d}</math>

==Further Information==
* Fitting models are described and derived as seen in [https://arxiv.org/abs/1109.6887v2 E. Mageson et al]. The coefficients derived are:
** <math>A_0</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
** <math>B_0</math> = Tr<math>[E_\psi \Lambda(\frac{\mathbb{1}}{d})]</math>
** <math>A_1</math> = Tr<math>[E_\psi \Lambda(\frac{Q_1\rho_\psi}{p} - \rho_\psi + \frac{(p-1)\mathbb{1}}{pd})]</math> + Tr<math>[E_\psi R_{m+1}(\frac{\rho_\psi}{p} - \frac{\mathbb{1}}{pd})]</math>
** <math>B_1</math> = Tr<math>[E_\psi R_{m+1}(\frac{\mathbb{1}}{d})]</math>
** <math>C_1</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
* The case where Randomized benchmarking fails: Suppose the noise is time dependent and for each <math>i, \Lambda_i = C_i^{\dagger}</math>. Then <math>F_g(m, \psi) = 1</math> for every <math>m</math> even though there is a substantial error on each <math>C_i</math> and so benchmarking fails.
* [https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.109.080505 Interleaved Randomized Benchmarking]: This protocol consists of interleaving random Clifford gates between the gate of interest and provides an estimate as well as theoretical bounds for the average error of the gate under test, so long as the average noise variation over all Clifford gates is small. Here the procedure followed is:
** Choose <math>K</math> sequences of Clifford elements where the first Clifford <math>C_{i_1}</math> in each sequence is chosen uniformly at random from Clif<math>_n</math>, the second is always chosen to be <math>C</math>(gate of interest), and alternate between uniformly random Clifford elements and deterministic <math>C</math> up to the <math>m^{th}</math> random gate.
** The <math>(m+1)^{th}</math> gate is chosen to be the inverse of the composition of the first <math>m</math> random gates and interlaced <math>C</math> gates.
** The rest of the steps remain the same and finally after plotting the new average sequence fidelity with the sequence length and fitting it into either the gate and time dependent or the gate and time independent model, we receive the new depolarizing parameter obtained is <math>p_c</math>, which replaces <math>p</math>.
** The new gate error is calculated as <math>r_c = \frac{(d-1)(1-p_c/p)}{d}</math>
* Wallman, Granade, Harper, F., NJP 2015 [[Purity benchmarking]]: A unitarity can be estimated via purity benchmarking, which is an RB-like experiment that estimates a decay rate.

==Related Papers==
* E.Knill et al (2007) arXiv:0707.0963: gate and time-independent noise model
* E. Mageson et al (2011) arXiv:1009.3639: multi-parameter model
* Magesan et al. PRL (2012): Interleaved Randomized Benchmarking
* Harper et al (2016) arXiv:1608.02943v2: Interleaved Randomised Benchmarking to estimate fidelity of T gates
* Wallman, Granade, Harper, F., NJP 2015: Purity benchmarking

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Standard Randomised Benchmarking

2020-03-24T10:50:35Z

Rhea:

[https://arxiv.org/abs/1109.6887v2 Randomized benchmarking] is a protocol that yields estimates of the computationally relevant errors without relying on accurate quantum state preparation and measurement. This is used to determine the error probability per gate in computational context and also gives an overall [[average fidelity]] for the noise in the gates.

'''Tags:''' [[:Category: Certification protocol|Certification Protocol]], [[:Category: Average gate fidelity|Average gate fidelity]], [[Randomised Benchmarking]], Clifford group

==Assumptions==
* The measurements performed are trusted.
* Noise model can be assumed to be gate and time-dependent or gate and time-independent.
* The noise model is independent and identically distributed (IID).

==Outline==
Randomized benchmarking method involves applying many random sequences of gates of varying lengths to a standard initial state. Each sequence ends with a randomized measurement that determines whether the correct final state was obtained. The average computationally relevant error per gate is obtained from the increase in error probability of the final measurements as a function of sequence length.

The random gates are taken from the [[Clifford group]]. The restriction to the Clifford group ensures that the measurements can be of one-qubit Pauli operators that yield at least one deterministic one-bit answer in the absence of errors.

This method consists of the following steps:

* A fixed sequence length is selected which is smaller than a predefined maximum sequence length. A random sequence of this length is chosen from the Clifford group.
* The operations are applied to the initial state corresponding to the selected sequence and then a final operator is applied which inverts all the previous operations.
* The final state is then measured to check if it matches the initial state. This process is performed several times with the same sequence to estimate the survival probability (the probability that the final state which returns to its initial state).
* Other random sequences of the same fixed sequence length are picked and the above-mentioned process is repeated to calculate the corresponding survival probability. This is used to calculate the average survival probability for the sequence length.
* The same procedure is repeated for multiple different sequence lengths.
* The observed survival probabilities are then plotted against the sequence length and then this is fit to an exponential decay curve, which is used to estimate the fidelity and also to calculate the average error rate which is the metric for randomized benchmarking.

==Hardware Requirements==
* Quantum computational resources to perform Clifford gates.
* Trusted Measurement device.

==Notation==
* <math>p</math>: Depolarizing parameter
* <math>d</math>: Dimension of Hilbert space
* <math>F_{avg}</math>: Average fidelity, <math>F_{avg} = p + \frac{1-p}{d}</math>
* <math>r</math>: Average error rate, <math>r = 1- F_{avg}, r = \frac{(d-1)(1-p)}{d}</math>
* <math>m</math>: Selected sequence length
* <math>K_m</math>: Total randomly selected sequence of <math>m</math> sequence length
* Clif<math>_n</math>: Clifford group
* C<math>_i</math>: Random element of Clifford group
* <math>S_{(i_1, ...,i_m)}</math> = <math>S_{\mathbf{i_m}}</math>: Random sequence of operations of length <math>m</math>
* <math>M</math>: Maximum sequence length of applying Clifford group Clif<math>_n</math>
* <math>\Lambda_{i,j}</math>: Implementation of C<math>_i</math> at time j (1 <math>\leq</math> j <math>\leq</math> M) results in this error map. <math>\Lambda_{i,1}, ..., \Lambda_{i,M}</math> are the different time-dependent noise operators affecting C<math>_i</math>.
* <math>|\psi\rangle</math>: initial state
* <math>E_{\psi}</math>: POVM element which takes into account the measurement error.
* <math>F_{seq}(m, \psi) = Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>: Survival probability of a sequence. <math>\rho_\psi</math> is a quantum state that takes into account errors in preparing <math>\langle \psi |\psi \rangle</math>
* <math>F_g^{(0)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time independent error model
* <math>F_g^{(1)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time dependent error model. In this model, the parameter <math>(q-p^2)</math> is a measure of the degree of gate-dependence in the error.
* <math>A_0, B_0</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time independent error model
* <math>A_1, B_1, C_1</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time dependent error model.
* <math>R_{m+1}</math>: <math>\frac{1}{|Clif_n|}\sum_i\Lambda_{i, m+1} \otimes (C_i \otimes \Lambda \otimes C_i^{\dagger})</math>

==Properties==
* '''Figure of merit''': average error rate, average gate fidelity
* The errors which are considered here are State preparation and measurement errors, error on the final gate, which are gate and time-independent errors. Gate and time-dependent errors can also be taken into consideration. This method is insensitive to SPAM error.
* The random gates are picked from the Clifford group. However in the case of [[interleaved randomized benchmarking]]
* For noise estimation, the uniform probability distribution over Clifford group comprises a [[unitary 2-design]].
* This protocol provides a scalable method for benchmarking the set of Clifford gates.
* To obtain a more accurate value for <math>p</math> one should always use the first order fitting model unless prior knowledge of the noise indicates that it is effectively gate-independent.

==Procedure Description==
'''Output''': Figure of merit: <math>r</math>

* For <math>m = 1, 2, ..., M-1</math>:
** For <math>k = 1, 2, ..., K_m</math> sequences:
*** For <math>j = 1, 2 ..., m+1</math>:
**** If <math>j == m+1</math>, apply inverse operator of previous operations
**** else, apply random operation C<math>_i</math>
*** Thus, <math>S_{\mathbf{i_m}} = \bigotimes^{m+1}_{j+1} (\Lambda_{(i_j, j)} C_i)</math> and <math>i_{m+1}</math> is uniquely determined by <math>(i_1, ...,i_m)</math>
*** Measure survival probability <math>Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>
** Estimate average survival probability <math>Tr[E_{\psi}S_{\mathbf{K_m}}(\rho_\psi)]</math> over all <math>K_m</math> sequences, where <math>S_{\mathbf{K_m}} = \frac{1}{K_m}\sum_{i_m} S_{i_m}</math>
* Fit the results for the averaged sequence fidelity for all <math>m</math> into the models:
** For gate and time independent error model:
*** <math>F_g^{(0)}(m, |\psi\rangle) = A_0p^m + B_0</math>
** For gate and time dependent error model:
*** <math>F_g^{(1)}(m, |\psi\rangle) = A_1p^m + B_1 + C_1(m-1)(q-p^2)p^{m-2}</math>
* <math>p</math> is extracted from the model and <math>r</math> is estimated, <math>r = \frac{(d-1)(1-p)}{d}</math>

==Further Information==
* Fitting models are described and derived as seen in [https://arxiv.org/abs/1109.6887v2 E. Mageson et al]. The coefficients derived are:
** <math>A_0</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
** <math>B_0</math> = Tr<math>[E_\psi \Lambda(\frac{\mathbb{1}}{d})]</math>
** <math>A_1</math> = Tr<math>[E_\psi \Lambda(\frac{Q_1\rho_\psi}{p} - \rho_\psi + \frac{(p-1)\mathbb{1}}{pd})]</math> + Tr<math>[E_\psi R_{m+1}(\frac{\rho_\psi}{p} - \frac{\mathbb{1}}{pd})]</math>
** <math>B_1</math> = Tr<math>[E_\psi R_{m+1}(\frac{\mathbb{1}}{d})]</math>
** <math>C_1</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
* The case where Randomized benchmarking fails: Suppose the noise is time dependent and for each <math>i, \Lambda_i = C_i^{\dagger}</math>. Then <math>F_g(m, \psi) = 1</math> for every <math>m</math> even though there is a substantial error on each <math>C_i</math> and so benchmarking fails.
* [https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.109.080505 Interleaved Randomized Benchmarking]: This protocol consists of interleaving random Clifford gates between the gate of interest and provides an estimate as well as theoretical bounds for the average error of the gate under test, so long as the average noise variation over all Clifford gates is small. Here the procedure followed is:
** Choose <math>K</math> sequences of Clifford elements where the first Clifford <math>C_{i_1}</math> in each sequence is chosen uniformly at random from Clif<math>_n</math>, the second is always chosen to be <math>C</math>(gate of interest), and alternate between uniformly random Clifford elements and deterministic <math>C</math> up to the <math>m^{th}</math> random gate.
** The <math>(m+1)^{th}</math> gate is chosen to be the inverse of the composition of the first <math>m</math> random gates and interlaced <math>C</math> gates.
** The rest of the steps remain the same and finally after plotting the new average sequence fidelity with the sequence length and fitting it into either the gate and time dependent or the gate and time independent model, we receive the new depolarizing parameter obtained is <math>p_c</math>, which replaces <math>p</math>.
** The new gate error is calculated as <math>r_c = \frac{(d-1)(1-p_c/p)}{d}</math>
* Wallman, Granade, Harper, F., NJP 2015 [[Purity benchmarking]]: A unitarity can be estimated via purity benchmarking, which is an RB-like experiment that estimates a decay rate.

==Related Papers==
* E.Knill et al (2007) arXiv:0707.0963: gate and time-independent noise model
* E. Mageson et al (2011) arXiv:1009.3639: multi-parameter model
* Magesan et al. PRL (2012): Interleaved Randomized Benchmarking
* Harper et al (2016) arXiv:1608.02943v2: Interleaved Randomised Benchmarking to estimate fidelity of T gates
* Wallman, Granade, Harper, F., NJP 2015: Purity benchmarking

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Standard Randomised Benchmarking

2020-03-24T10:20:59Z

Rhea: /* Further Information */

[https://arxiv.org/abs/1109.6887v2 Randomized benchmarking] is a protocol that yields estimates of the computationally relevant errors without relying on accurate quantum state preparation and measurement. This is used to determine the error probability per gate in computational context and also gives an overall [[average fidelity]] for the noise in the gates.

'''Tags:''' [[:Category: Certification protocol|Certification Protocol]], [[:Category: Average gate fidelity|Average gate fidelity]], Clifford group

==Assumptions==
* The measurements performed are trusted.
* Noise model can be assumed to be gate and time-dependent or gate and time-independent.
* The noise model is independent and identically distributed (IID).

==Outline==
Randomized benchmarking method involves applying many random sequences of gates of varying lengths to a standard initial state. Each sequence ends with a randomized measurement that determines whether the correct final state was obtained. The average computationally relevant error per gate is obtained from the increase in error probability of the final measurements as a function of sequence length.

The random gates are taken from the [[Clifford group]]. The restriction to the Clifford group ensures that the measurements can be of one-qubit Pauli operators that yield at least one deterministic one-bit answer in the absence of errors.

This method consists of the following steps:

* A fixed sequence length is selected which is smaller than a predefined maximum sequence length. A random sequence of this length is chosen from the Clifford group.
* The operations are applied to the initial state corresponding to the selected sequence and then a final operator is applied which inverts all the previous operations.
* The final state is then measured to check if it matches the initial state. This process is performed several times with the same sequence to estimate the survival probability (the probability that the final state which returns to its initial state).
* Other random sequences of the same fixed sequence length are picked and the above-mentioned process is repeated to calculate the corresponding survival probability. This is used to calculate the average survival probability for the sequence length.
* The same procedure is repeated for multiple different sequence lengths.
* The observed survival probabilities are then plotted against the sequence length and then this is fit to an exponential decay curve, which is used to estimate the fidelity and also to calculate the average error rate which is the metric for randomized benchmarking.

==Hardware Requirements==
* Quantum computational resources to perform Clifford gates.
* Trusted Measurement device.

==Notation==
* <math>p</math>: Depolarizing parameter
* <math>d</math>: Dimension of Hilbert space
* <math>F_{avg}</math>: Average fidelity, <math>F_{avg} = p + \frac{1-p}{d}</math>
* <math>r</math>: Average error rate, <math>r = 1- F_{avg}, r = \frac{(d-1)(1-p)}{d}</math>
* <math>m</math>: Selected sequence length
* <math>K_m</math>: Total randomly selected sequence of <math>m</math> sequence length
* Clif<math>_n</math>: Clifford group
* C<math>_i</math>: Random element of Clifford group
* <math>S_{(i_1, ...,i_m)}</math> = <math>S_{\mathbf{i_m}}</math>: Random sequence of operations of length <math>m</math>
* <math>M</math>: Maximum sequence length of applying Clifford group Clif<math>_n</math>
* <math>\Lambda_{i,j}</math>: Implementation of C<math>_i</math> at time j (1 <math>\leq</math> j <math>\leq</math> M) results in this error map. <math>\Lambda_{i,1}, ..., \Lambda_{i,M}</math> are the different time-dependent noise operators affecting C<math>_i</math>.
* <math>|\psi\rangle</math>: initial state
* <math>E_{\psi}</math>: POVM element which takes into account the measurement error.
* <math>F_{seq}(m, \psi) = Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>: Survival probability of a sequence. <math>\rho_\psi</math> is a quantum state that takes into account errors in preparing <math>\langle \psi |\psi \rangle</math>
* <math>F_g^{(0)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time independent error model
* <math>F_g^{(1)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time dependent error model. In this model, the parameter <math>(q-p^2)</math> is a measure of the degree of gate-dependence in the error.
* <math>A_0, B_0</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time independent error model
* <math>A_1, B_1, C_1</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time dependent error model.
* <math>R_{m+1}</math>: <math>\frac{1}{|Clif_n|}\sum_i\Lambda_{i, m+1} \otimes (C_i \otimes \Lambda \otimes C_i^{\dagger})</math>

==Properties==
* '''Figure of merit''': average error rate, average gate fidelity
* The errors which are considered here are State preparation and measurement errors, error on the final gate, which are gate and time-independent errors. Gate and time-dependent errors can also be taken into consideration. This method is insensitive to SPAM error.
* The random gates are picked from the Clifford group. However in the case of [[interleaved randomized benchmarking]]
* For noise estimation, the uniform probability distribution over Clifford group comprises a [[unitary 2-design]].
* This protocol provides a scalable method for benchmarking the set of Clifford gates.
* To obtain a more accurate value for <math>p</math> one should always use the first order fitting model unless prior knowledge of the noise indicates that it is effectively gate-independent.

==Procedure Description==
'''Output''': Figure of merit: <math>r</math>

* For <math>m = 1, 2, ..., M-1</math>:
** For <math>k = 1, 2, ..., K_m</math> sequences:
*** For <math>j = 1, 2 ..., m+1</math>:
**** If <math>j == m+1</math>, apply inverse operator of previous operations
**** else, apply random operation C<math>_i</math>
*** Thus, <math>S_{\mathbf{i_m}} = \bigotimes^{m+1}_{j+1} (\Lambda_{(i_j, j)} C_i)</math> and <math>i_{m+1}</math> is uniquely determined by <math>(i_1, ...,i_m)</math>
*** Measure survival probability <math>Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>
** Estimate average survival probability <math>Tr[E_{\psi}S_{\mathbf{K_m}}(\rho_\psi)]</math> over all <math>K_m</math> sequences, where <math>S_{\mathbf{K_m}} = \frac{1}{K_m}\sum_{i_m} S_{i_m}</math>
* Fit the results for the averaged sequence fidelity for all <math>m</math> into the models:
** For gate and time independent error model:
*** <math>F_g^{(0)}(m, |\psi\rangle) = A_0p^m + B_0</math>
** For gate and time dependent error model:
*** <math>F_g^{(1)}(m, |\psi\rangle) = A_1p^m + B_1 + C_1(m-1)(q-p^2)p^{m-2}</math>
* <math>p</math> is extracted from the model and <math>r</math> is estimated, <math>r = \frac{(d-1)(1-p)}{d}</math>

==Further Information==
* Fitting models are described and derived as seen in [https://arxiv.org/abs/1109.6887v2 E. Mageson et al]. The coefficients derived are:
** <math>A_0</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
** <math>B_0</math> = Tr<math>[E_\psi \Lambda(\frac{\mathbb{1}}{d})]</math>
** <math>A_1</math> = Tr<math>[E_\psi \Lambda(\frac{Q_1\rho_\psi}{p} - \rho_\psi + \frac{(p-1)\mathbb{1}}{pd})]</math> + Tr<math>[E_\psi R_{m+1}(\frac{\rho_\psi}{p} - \frac{\mathbb{1}}{pd})]</math>
** <math>B_1</math> = Tr<math>[E_\psi R_{m+1}(\frac{\mathbb{1}}{d})]</math>
** <math>C_1</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
* The case where Randomized benchmarking fails: Suppose the noise is time dependent and for each <math>i, \Lambda_i = C_i^{\dagger}</math>. Then <math>F_g(m, \psi) = 1</math> for every <math>m</math> even though there is a substantial error on each <math>C_i</math> and so benchmarking fails.
* [https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.109.080505 Interleaved Randomized Benchmarking]: This protocol consists of interleaving random Clifford gates between the gate of interest and provides an estimate as well as theoretical bounds for the average error of the gate under test, so long as the average noise variation over all Clifford gates is small. Here the procedure followed is:
** Choose <math>K</math> sequences of Clifford elements where the first Clifford <math>C_{i_1}</math> in each sequence is chosen uniformly at random from Clif<math>_n</math>, the second is always chosen to be <math>C</math>(gate of interest), and alternate between uniformly random Clifford elements and deterministic <math>C</math> up to the <math>m^{th}</math> random gate.
** The <math>(m+1)^{th}</math> gate is chosen to be the inverse of the composition of the first <math>m</math> random gates and interlaced <math>C</math> gates.
** The rest of the steps remain the same and finally after plotting the new average sequence fidelity with the sequence length and fitting it into either the gate and time dependent or the gate and time independent model, we receive the new depolarizing parameter obtained is <math>p_c</math>, which replaces <math>p</math>.
** The new gate error is calculated as <math>r_c = \frac{(d-1)(1-p_c/p)}{d}</math>
* Wallman, Granade, Harper, F., NJP 2015 [[Purity benchmarking]]: A unitarity can be estimated via purity benchmarking, which is an RB-like experiment that estimates a decay rate.

==Related Papers==
* E.Knill et al (2007) arXiv:0707.0963: gate and time-independent noise model
* E. Mageson et al (2011) arXiv:1009.3639: multi-parameter model
* Magesan et al. PRL (2012): Interleaved Randomized Benchmarking
* Harper et al (2016) arXiv:1608.02943v2: Interleaved Randomised Benchmarking to estimate fidelity of T gates
* Wallman, Granade, Harper, F., NJP 2015: Purity benchmarking

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Standard Randomised Benchmarking

2020-03-24T10:20:20Z

Rhea:

[https://arxiv.org/abs/1109.6887v2 Randomized benchmarking] is a protocol that yields estimates of the computationally relevant errors without relying on accurate quantum state preparation and measurement. This is used to determine the error probability per gate in computational context and also gives an overall [[average fidelity]] for the noise in the gates.

'''Tags:''' [[:Category: Certification protocol|Certification Protocol]], [[:Category: Average gate fidelity|Average gate fidelity]], Clifford group

==Assumptions==
* The measurements performed are trusted.
* Noise model can be assumed to be gate and time-dependent or gate and time-independent.
* The noise model is independent and identically distributed (IID).

==Outline==
Randomized benchmarking method involves applying many random sequences of gates of varying lengths to a standard initial state. Each sequence ends with a randomized measurement that determines whether the correct final state was obtained. The average computationally relevant error per gate is obtained from the increase in error probability of the final measurements as a function of sequence length.

The random gates are taken from the [[Clifford group]]. The restriction to the Clifford group ensures that the measurements can be of one-qubit Pauli operators that yield at least one deterministic one-bit answer in the absence of errors.

This method consists of the following steps:

* A fixed sequence length is selected which is smaller than a predefined maximum sequence length. A random sequence of this length is chosen from the Clifford group.
* The operations are applied to the initial state corresponding to the selected sequence and then a final operator is applied which inverts all the previous operations.
* The final state is then measured to check if it matches the initial state. This process is performed several times with the same sequence to estimate the survival probability (the probability that the final state which returns to its initial state).
* Other random sequences of the same fixed sequence length are picked and the above-mentioned process is repeated to calculate the corresponding survival probability. This is used to calculate the average survival probability for the sequence length.
* The same procedure is repeated for multiple different sequence lengths.
* The observed survival probabilities are then plotted against the sequence length and then this is fit to an exponential decay curve, which is used to estimate the fidelity and also to calculate the average error rate which is the metric for randomized benchmarking.

==Hardware Requirements==
* Quantum computational resources to perform Clifford gates.
* Trusted Measurement device.

==Notation==
* <math>p</math>: Depolarizing parameter
* <math>d</math>: Dimension of Hilbert space
* <math>F_{avg}</math>: Average fidelity, <math>F_{avg} = p + \frac{1-p}{d}</math>
* <math>r</math>: Average error rate, <math>r = 1- F_{avg}, r = \frac{(d-1)(1-p)}{d}</math>
* <math>m</math>: Selected sequence length
* <math>K_m</math>: Total randomly selected sequence of <math>m</math> sequence length
* Clif<math>_n</math>: Clifford group
* C<math>_i</math>: Random element of Clifford group
* <math>S_{(i_1, ...,i_m)}</math> = <math>S_{\mathbf{i_m}}</math>: Random sequence of operations of length <math>m</math>
* <math>M</math>: Maximum sequence length of applying Clifford group Clif<math>_n</math>
* <math>\Lambda_{i,j}</math>: Implementation of C<math>_i</math> at time j (1 <math>\leq</math> j <math>\leq</math> M) results in this error map. <math>\Lambda_{i,1}, ..., \Lambda_{i,M}</math> are the different time-dependent noise operators affecting C<math>_i</math>.
* <math>|\psi\rangle</math>: initial state
* <math>E_{\psi}</math>: POVM element which takes into account the measurement error.
* <math>F_{seq}(m, \psi) = Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>: Survival probability of a sequence. <math>\rho_\psi</math> is a quantum state that takes into account errors in preparing <math>\langle \psi |\psi \rangle</math>
* <math>F_g^{(0)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time independent error model
* <math>F_g^{(1)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time dependent error model. In this model, the parameter <math>(q-p^2)</math> is a measure of the degree of gate-dependence in the error.
* <math>A_0, B_0</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time independent error model
* <math>A_1, B_1, C_1</math>: Coefficients that absorb the state preparation and measurement errors as well as the error on the final gate for gate and time dependent error model.
* <math>R_{m+1}</math>: <math>\frac{1}{|Clif_n|}\sum_i\Lambda_{i, m+1} \otimes (C_i \otimes \Lambda \otimes C_i^{\dagger})</math>

==Properties==
* '''Figure of merit''': average error rate, average gate fidelity
* The errors which are considered here are State preparation and measurement errors, error on the final gate, which are gate and time-independent errors. Gate and time-dependent errors can also be taken into consideration. This method is insensitive to SPAM error.
* The random gates are picked from the Clifford group. However in the case of [[interleaved randomized benchmarking]]
* For noise estimation, the uniform probability distribution over Clifford group comprises a [[unitary 2-design]].
* This protocol provides a scalable method for benchmarking the set of Clifford gates.
* To obtain a more accurate value for <math>p</math> one should always use the first order fitting model unless prior knowledge of the noise indicates that it is effectively gate-independent.

==Procedure Description==
'''Output''': Figure of merit: <math>r</math>

* For <math>m = 1, 2, ..., M-1</math>:
** For <math>k = 1, 2, ..., K_m</math> sequences:
*** For <math>j = 1, 2 ..., m+1</math>:
**** If <math>j == m+1</math>, apply inverse operator of previous operations
**** else, apply random operation C<math>_i</math>
*** Thus, <math>S_{\mathbf{i_m}} = \bigotimes^{m+1}_{j+1} (\Lambda_{(i_j, j)} C_i)</math> and <math>i_{m+1}</math> is uniquely determined by <math>(i_1, ...,i_m)</math>
*** Measure survival probability <math>Tr[E_{\psi}S_{\mathbf{i_m}}(\rho_\psi)]</math>
** Estimate average survival probability <math>Tr[E_{\psi}S_{\mathbf{K_m}}(\rho_\psi)]</math> over all <math>K_m</math> sequences, where <math>S_{\mathbf{K_m}} = \frac{1}{K_m}\sum_{i_m} S_{i_m}</math>
* Fit the results for the averaged sequence fidelity for all <math>m</math> into the models:
** For gate and time independent error model:
*** <math>F_g^{(0)}(m, |\psi\rangle) = A_0p^m + B_0</math>
** For gate and time dependent error model:
*** <math>F_g^{(1)}(m, |\psi\rangle) = A_1p^m + B_1 + C_1(m-1)(q-p^2)p^{m-2}</math>
* <math>p</math> is extracted from the model and <math>r</math> is estimated, <math>r = \frac{(d-1)(1-p)}{d}</math>

==Further Information==
* Fitting models are described and derived as seen in [https://arxiv.org/abs/1109.6887v2 E. Mageson et al]. The coefficients derived are:
** <math>A_0</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
** <math>B_0</math> = Tr<math>[E_\psi \Lambda(\frac{\mathbb{1}}{d})]</math>
** <math>A_1</math> = Tr<math>[E_\psi \Lambda(\frac{Q_1\rho_\psi}{p} - \rho_\psi + \frac{(p-1)\mathbb{1}}{pd})]</math> + Tr<math>[E_\psi R_{m+1}(\frac{\rho_\psi}{p} - \frac{\mathbb{1}}{pd})]</math>
** <math>B_1</math> = Tr<math>[E_\psi R_{m+1}(\frac{\mathbb{1}}{d})]</math>
** <math>C_1</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
* The case where Randomized benchmarking fails: Suppose the noise is time dependent and for each <math>i, \Lambda_i = C_i^{\dagger}</math>. Then <math>F_g(m, \psi) = 1</math> for every <math>m</math> even though there is a substantial error on each <math>C_i</math> and so benchmarking fails.
* [https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.109.080505 Interleaved Randomized Benchmarking]: This protocol consists of interleaving random Clifford gates between the gate of interest and provides an estimate as well as theoretical bounds for the average error of the gate under test, so long as the average noise variation over all Clifford gates is small. Here the procedure followed is:
** Choose <math>K</math> sequences of Clifford elements where the first Clifford <math>C_{i_1}</math> in each sequence is chosen uniformly at random from Clif$_n$, the second is always chosen to be <math>C</math>(gate of interest), and alternate between uniformly random Clifford elements and deterministic <math>C</math> up to the <math>m^{th}</math> random gate.
** The <math>(m+1)^{th}</math> gate is chosen to be the inverse of the composition of the first <math>m</math> random gates and interlaced <math>C</math> gates.
** The rest of the steps remain the same and finally after plotting the new average sequence fidelity with the sequence length and fitting it into either the gate and time dependent or the gate and time independent model, we receive the new depolarizing parameter obtained is <math>p_c</math>, which replaces <math>p</math>.
** The new gate error is calculated as <math>r_c = \frac{(d-1)(1-p_c/p)}{d}</math>
* Wallman, Granade, Harper, F., NJP 2015 [[Purity benchmarking]]: A unitarity can be estimated via purity benchmarking, which is an RB-like experiment that estimates a decay rate.

==Related Papers==
* E.Knill et al (2007) arXiv:0707.0963: gate and time-independent noise model
* E. Mageson et al (2011) arXiv:1009.3639: multi-parameter model
* Magesan et al. PRL (2012): Interleaved Randomized Benchmarking
* Harper et al (2016) arXiv:1608.02943v2: Interleaved Randomised Benchmarking to estimate fidelity of T gates
* Wallman, Granade, Harper, F., NJP 2015: Purity benchmarking

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Quantum Volume Estimation

2020-03-23T20:09:38Z

Rhea: /* Properties */

[https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.032328 Quantum Volume (QV)] is a single-number metric that can be measured using a concrete protocol on near-term quantum computers of modest size. The QV method quantifies the largest random circuit of equal width and depth that the computer successfully implements. The quantum volume protocol is strongly related to gate error rate and is influenced by underlying qubit connectivity and gate parallelism. This protocol is based on the performance of random model circuits with a fixed but generic form.

'''Tags:''' [[Certification protocol]], [[Quantum Volume estimation protocol]]

==Assumptions==

* The transpiler is free to use all available tricks and hardware resources to implement the model circuit.
* The transpiler should make an honest attempt to implement the model circuit, and not merely choose a relatively simple operation far from the model circuit that nevertheless produces the heavy outputs for it.

==Outline==
The quantum volume protocol is strongly related to gate error rate and is influenced by underlying qubit connectivity and gate parallelism. This protocol is based on the performance of random model circuits with a fixed but generic form.

A model circuit is consists of <math>d</math> layers of random permutations of the <math>m</math> different qubit labels, followed by random two-qubit gates. When the circuit width m is odd, one of the qubits is idle in each layer. Each two-qubit gate used in the previous step is sampled from the [[Haar measure on SU(4)]].

Heavy output generation problem is used to define if the model circuit mentioned above is fully implemented in practice. From the outputs of all the implementations of the model circuit, we get an ideal output distribution. From this, we get the set of output probabilities and we can obtain the median of this set. The heavy outputs are the outputs for which the output probability will be greater than the median of the set of probabilities. The heavy output generation problem is to produce a
set of output strings such that more than two-thirds are
heavy.

To evaluate heavy output generation, we implement model circuits using the gate set provided by the target system, using the available hardware. For this purpose, a quantum circuit-to-circuit transpiler is used, which finds an implementation of the model circuit, where the approximation error.

This method to compute the quantum volume of a device consists of the following steps:

* The Quantum transpiler tries to implement the model circuit such that the approximation error is limited. From here, we get the distribution for the implementation of the model circuit, which we use to calculate the probability of sampling a heavy output.
* The heavy outputs are also computed using the ideal output distribution of the model circuit.
* The probability of observing a heavy output by implementing a randomly selected depth <math>d</math> model circuit is also computed using the probability of sampling a heavy output computed in the step above.
* We define the achievable depth <math>d(m)</math> to be the largest <math>d</math> such that we are confident that the probability of observing a heavy output is greater than <math>2/3</math> (as the heavy output generation problem is to produce a set of output strings such that more than two-thirds are heavy.)
* The data of achievable depth is gathered by sweeping over values of width <math>m</math> and depth <math>d</math> of the model circuit.
* Using all the data gathered, the quantum volume is computed. The quantum volume treats the width and depth of a model circuit with equal importance and measures the largest square shaped (i.e., <math>m = d</math>) model circuit a quantum computer can implement successfully on average.

==Notation==

* <math>U</math>: Model circuit
* <math>U'</math>: Implementation of the model circuit by the quantum transpiler
* <math>m</math>: width of the model circuit
* <math>d</math>: depth of the model circuit
* <math>F_{avg}(U, U')</math>: Average fidelity between <math>U</math> and <math>U'</math>
* <math>\epsilon</math>: approximation error
* <math>d(m)</math>: Achievable depth, which is the largest <math>d</math> such that we are confident that the probability of observing a heavy output is greater than <math>2/3</math>
* <math>V_Q</math>: Quantum Volume
* <math>H_U</math>: Set of heavy outputs for a model circuit <math>U</math>
* <math>x</math>: Outcome of executing <math>U'</math>, which is a observable bit string, <math>x \in \{0,1\}^m</math>
* <math>p_U(x)</math>: Ideal output distribution for <math>U</math>. <math>p_U(x) = |\langle x|U|0\rangle|^2</math>
* <math>p_{med}</math>: median of the set of probabilities
* <math>n_c</math>: Number of repetitions, <math>n_c>100</math>
* <math>n_s</math>: Number of repetitions

==Hardware Requirements==

* Quantum Computing device with a gate set
* Measurement device

==Properties==

* '''Figure of merit''': Quantum Volume
* Quantum computing systems with high-fidelity operations, high connectivity, large calibrated gate sets, and circuit rewriting tool chains are expected to have higher quantum volumes
* The protocol can be implemented with any universal programmable quantum computing device. Quantum volume is architecture-independent, and can be applied to any system that is capable of running quantum circuits.
* The method used to compute the heavy outputs from the ideal output distribution of the model circuit scales exponentially with the width <math>m</math>.
* Ideally, the probability of observing a heavy output would be estimated using all of the qubits of a large device, but NISQ devices have appreciable error rates, so we begin with small model circuits and progress to larger ones.
* The quantum volume treats the width and depth of a model circuit with equal importance and measures the largest square shaped (i.e., <math>m = d</math>) model circuit a quantum computer can implement successfully on average.
* Given a model circuit <math>U</math>, a circuit-to-circuit transpiler finds an implementation <math>U'</math> for the target system such that <math>1- F_{avg}(U, U') \leq \epsilon \ll 1</math>
* There are two possible paths for increasing the quantum volume, which is given by the numerical simulations for given connectivity. The first path is to prioritize improving the gate fidelity above other operations. This
sets the roadmap for device performance to focus on the errors that limit gate performance, such as coherence and
calibration errors. The second path stems from the observation that, for these devices and this metric, circuit
optimization is becoming important.

==Protocol Description==

'''Function''': ComputeHeavyOutputs<math>(U, m)</math>

'''Input''': <math>U, m</math>

'''Output''': <math>H_U</math>

* Obtain <math>p_U(x)</math> for <math>x \in \{0,1\}^m</math>
* Sort in ascending order <math>p_0 \leq p_1 ... \leq p_{2^m -1}</math>
* <math>p_{med} = (p_{2^{m-1}} + p_{2^{m-1}-1})/2 </math>
* <math>H_U = \{x\in \{0,1\}^m</math> such that <math>p_U(x) > p_{med}\}</math>

'''Function''': ComputeQuantumVolume

'''Output''': Figure of merit: Quantum Volume, <math>V_Q</math>

* For <math>i = 1, 2, ..., m</math>:
** For <math>j = 1, 2, ..., d</math>:
*** <math>d(m) = 0</math>
*** <math>n_h = 0</math>
*** For <math>k = 1, 2, ..., n_c</math>:
**** Pick random model circuit <math>U</math>
**** <math>H_U =</math> ComputeHeavyOutputs<math>(U, m)</math>
**** Compile <math>U'</math>
**** For <math>l = 1, 2, ..., n_s</math>:
***** Get output <math>x</math>
***** If <math>x\in H_U</math> then <math>n_h = n_h + 1</math>
*** If <math>\frac{n_h-2\sqrt{n_h(n_s-n_h/n_c)}}{n_c n_S} > \frac{2}{3}</math>
**** <math>d(m) = </math>max<math>(d(m), d)</math>
**** Store data <math>(m, d(m))</math>
* Calculate <math>V_Q</math> from stored data, where log<math>_2 V_Q</math> = argmax<math>_m</math> min<math>(m, d(m))</math>

==Further Information==

== Related Papers ==
* Andrew W. Cross et al (2011) arXiv:1811.12926v3: Validating quantum computers using randomized model circuits

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Quantum Volume Estimation

2020-03-23T19:58:50Z

Rhea: /* Properties */

[https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.032328 Quantum Volume (QV)] is a single-number metric that can be measured using a concrete protocol on near-term quantum computers of modest size. The QV method quantifies the largest random circuit of equal width and depth that the computer successfully implements. The quantum volume protocol is strongly related to gate error rate and is influenced by underlying qubit connectivity and gate parallelism. This protocol is based on the performance of random model circuits with a fixed but generic form.

'''Tags:''' [[Certification protocol]], [[Quantum Volume estimation protocol]]

==Assumptions==

* The transpiler is free to use all available tricks and hardware resources to implement the model circuit.
* The transpiler should make an honest attempt to implement the model circuit, and not merely choose a relatively simple operation far from the model circuit that nevertheless produces the heavy outputs for it.

==Outline==
The quantum volume protocol is strongly related to gate error rate and is influenced by underlying qubit connectivity and gate parallelism. This protocol is based on the performance of random model circuits with a fixed but generic form.

A model circuit is consists of <math>d</math> layers of random permutations of the <math>m</math> different qubit labels, followed by random two-qubit gates. When the circuit width m is odd, one of the qubits is idle in each layer. Each two-qubit gate used in the previous step is sampled from the [[Haar measure on SU(4)]].

Heavy output generation problem is used to define if the model circuit mentioned above is fully implemented in practice. From the outputs of all the implementations of the model circuit, we get an ideal output distribution. From this, we get the set of output probabilities and we can obtain the median of this set. The heavy outputs are the outputs for which the output probability will be greater than the median of the set of probabilities. The heavy output generation problem is to produce a
set of output strings such that more than two-thirds are
heavy.

To evaluate heavy output generation, we implement model circuits using the gate set provided by the target system, using the available hardware. For this purpose, a quantum circuit-to-circuit transpiler is used, which finds an implementation of the model circuit, where the approximation error.

This method to compute the quantum volume of a device consists of the following steps:

* The Quantum transpiler tries to implement the model circuit such that the approximation error is limited. From here, we get the distribution for the implementation of the model circuit, which we use to calculate the probability of sampling a heavy output.
* The heavy outputs are also computed using the ideal output distribution of the model circuit.
* The probability of observing a heavy output by implementing a randomly selected depth <math>d</math> model circuit is also computed using the probability of sampling a heavy output computed in the step above.
* We define the achievable depth <math>d(m)</math> to be the largest <math>d</math> such that we are confident that the probability of observing a heavy output is greater than <math>2/3</math> (as the heavy output generation problem is to produce a set of output strings such that more than two-thirds are heavy.)
* The data of achievable depth is gathered by sweeping over values of width <math>m</math> and depth <math>d</math> of the model circuit.
* Using all the data gathered, the quantum volume is computed. The quantum volume treats the width and depth of a model circuit with equal importance and measures the largest square shaped (i.e., <math>m = d</math>) model circuit a quantum computer can implement successfully on average.

==Notation==

* <math>U</math>: Model circuit
* <math>U'</math>: Implementation of the model circuit by the quantum transpiler
* <math>m</math>: width of the model circuit
* <math>d</math>: depth of the model circuit
* <math>F_{avg}(U, U')</math>: Average fidelity between <math>U</math> and <math>U'</math>
* <math>\epsilon</math>: approximation error
* <math>d(m)</math>: Achievable depth, which is the largest <math>d</math> such that we are confident that the probability of observing a heavy output is greater than <math>2/3</math>
* <math>V_Q</math>: Quantum Volume
* <math>H_U</math>: Set of heavy outputs for a model circuit <math>U</math>
* <math>x</math>: Outcome of executing <math>U'</math>, which is a observable bit string, <math>x \in \{0,1\}^m</math>
* <math>p_U(x)</math>: Ideal output distribution for <math>U</math>. <math>p_U(x) = |\langle x|U|0\rangle|^2</math>
* <math>p_{med}</math>: median of the set of probabilities
* <math>n_c</math>: Number of repetitions, <math>n_c>100</math>
* <math>n_s</math>: Number of repetitions

==Hardware Requirements==

* Quantum Computing device with a gate set
* Measurement device

==Properties==

* '''Figure of merit''': Quantum Volume
* Quantum computing systems with high-fidelity operations, high connectivity, large calibrated gate sets, and circuit rewriting tool chains are expected to have higher quantum volumes
* The protocol can be implemented with any universal programmable quantum computing device. Quantum volume is architecture-independent, and can be applied to any system that is capable of running quantum circuits.
* The method used to compute the heavy outputs from the ideal output distribution of the model circuit scales exponentially with the width <math>m</math>.
* Ideally, the probability of observing a heavy output would be estimated using all of the qubits of a large device, but NISQ devices have appreciable error rates, so we begin with small model circuits and progress to larger ones.
* The quantum volume treats the width and depth of a model circuit with equal importance and measures the largest square shaped (i.e., <math>m = d</math>) model circuit a quantum computer can implement successfully on average.
* Given a model circuit <math>U</math>, a circuit-to-circuit transpiler finds an implementation <math>U'</math> for the target system such that <math>1- F_{avg}(U, U') \leq \epsilon \ll 1</math>

==Protocol Description==

'''Function''': ComputeHeavyOutputs<math>(U, m)</math>

'''Input''': <math>U, m</math>

'''Output''': <math>H_U</math>

* Obtain <math>p_U(x)</math> for <math>x \in \{0,1\}^m</math>
* Sort in ascending order <math>p_0 \leq p_1 ... \leq p_{2^m -1}</math>
* <math>p_{med} = (p_{2^{m-1}} + p_{2^{m-1}-1})/2 </math>
* <math>H_U = \{x\in \{0,1\}^m</math> such that <math>p_U(x) > p_{med}\}</math>

'''Function''': ComputeQuantumVolume

'''Output''': Figure of merit: Quantum Volume, <math>V_Q</math>

* For <math>i = 1, 2, ..., m</math>:
** For <math>j = 1, 2, ..., d</math>:
*** <math>d(m) = 0</math>
*** <math>n_h = 0</math>
*** For <math>k = 1, 2, ..., n_c</math>:
**** Pick random model circuit <math>U</math>
**** <math>H_U =</math> ComputeHeavyOutputs<math>(U, m)</math>
**** Compile <math>U'</math>
**** For <math>l = 1, 2, ..., n_s</math>:
***** Get output <math>x</math>
***** If <math>x\in H_U</math> then <math>n_h = n_h + 1</math>
*** If <math>\frac{n_h-2\sqrt{n_h(n_s-n_h/n_c)}}{n_c n_S} > \frac{2}{3}</math>
**** <math>d(m) = </math>max<math>(d(m), d)</math>
**** Store data <math>(m, d(m))</math>
* Calculate <math>V_Q</math> from stored data, where log<math>_2 V_Q</math> = argmax<math>_m</math> min<math>(m, d(m))</math>

==Further Information==

== Related Papers ==
* Andrew W. Cross et al (2011) arXiv:1811.12926v3: Validating quantum computers using randomized model circuits

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Quantum Volume Estimation

2020-03-23T19:58:07Z

Rhea: /* Properties */

[https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.032328 Quantum Volume (QV)] is a single-number metric that can be measured using a concrete protocol on near-term quantum computers of modest size. The QV method quantifies the largest random circuit of equal width and depth that the computer successfully implements. The quantum volume protocol is strongly related to gate error rate and is influenced by underlying qubit connectivity and gate parallelism. This protocol is based on the performance of random model circuits with a fixed but generic form.

'''Tags:''' [[Certification protocol]], [[Quantum Volume estimation protocol]]

==Assumptions==

* The transpiler is free to use all available tricks and hardware resources to implement the model circuit.
* The transpiler should make an honest attempt to implement the model circuit, and not merely choose a relatively simple operation far from the model circuit that nevertheless produces the heavy outputs for it.

==Outline==
The quantum volume protocol is strongly related to gate error rate and is influenced by underlying qubit connectivity and gate parallelism. This protocol is based on the performance of random model circuits with a fixed but generic form.

A model circuit is consists of <math>d</math> layers of random permutations of the <math>m</math> different qubit labels, followed by random two-qubit gates. When the circuit width m is odd, one of the qubits is idle in each layer. Each two-qubit gate used in the previous step is sampled from the [[Haar measure on SU(4)]].

Heavy output generation problem is used to define if the model circuit mentioned above is fully implemented in practice. From the outputs of all the implementations of the model circuit, we get an ideal output distribution. From this, we get the set of output probabilities and we can obtain the median of this set. The heavy outputs are the outputs for which the output probability will be greater than the median of the set of probabilities. The heavy output generation problem is to produce a
set of output strings such that more than two-thirds are
heavy.

To evaluate heavy output generation, we implement model circuits using the gate set provided by the target system, using the available hardware. For this purpose, a quantum circuit-to-circuit transpiler is used, which finds an implementation of the model circuit, where the approximation error.

This method to compute the quantum volume of a device consists of the following steps:

* The Quantum transpiler tries to implement the model circuit such that the approximation error is limited. From here, we get the distribution for the implementation of the model circuit, which we use to calculate the probability of sampling a heavy output.
* The heavy outputs are also computed using the ideal output distribution of the model circuit.
* The probability of observing a heavy output by implementing a randomly selected depth <math>d</math> model circuit is also computed using the probability of sampling a heavy output computed in the step above.
* We define the achievable depth <math>d(m)</math> to be the largest <math>d</math> such that we are confident that the probability of observing a heavy output is greater than <math>2/3</math> (as the heavy output generation problem is to produce a set of output strings such that more than two-thirds are heavy.)
* The data of achievable depth is gathered by sweeping over values of width <math>m</math> and depth <math>d</math> of the model circuit.
* Using all the data gathered, the quantum volume is computed. The quantum volume treats the width and depth of a model circuit with equal importance and measures the largest square shaped (i.e., <math>m = d</math>) model circuit a quantum computer can implement successfully on average.

==Notation==

* <math>U</math>: Model circuit
* <math>U'</math>: Implementation of the model circuit by the quantum transpiler
* <math>m</math>: width of the model circuit
* <math>d</math>: depth of the model circuit
* <math>F_{avg}(U, U')</math>: Average fidelity between <math>U</math> and <math>U'</math>
* <math>\epsilon</math>: approximation error
* <math>d(m)</math>: Achievable depth, which is the largest <math>d</math> such that we are confident that the probability of observing a heavy output is greater than <math>2/3</math>
* <math>V_Q</math>: Quantum Volume
* <math>H_U</math>: Set of heavy outputs for a model circuit <math>U</math>
* <math>x</math>: Outcome of executing <math>U'</math>, which is a observable bit string, <math>x \in \{0,1\}^m</math>
* <math>p_U(x)</math>: Ideal output distribution for <math>U</math>. <math>p_U(x) = |\langle x|U|0\rangle|^2</math>
* <math>p_{med}</math>: median of the set of probabilities
* <math>n_c</math>: Number of repetitions, <math>n_c>100</math>
* <math>n_s</math>: Number of repetitions

==Hardware Requirements==

* Quantum Computing device with a gate set
* Measurement device

==Properties==

* '''Figure of merit''': Quantum Volume
* Quantum computing systems with high-fidelity operations, high connectivity, large calibrated gate sets, and circuit rewriting tool chains are expected to have higher quantum volumes
* The protocol can be implemented with any universal programmable quantum computing device. Quantum volume is
architecture-independent, and can be applied to any system that is capable of running quantum circuits.
* The method used to compute the heavy outputs from the ideal output distribution of the model circuit scales exponentially with the width <math>m</math>.
* Ideally, the probability of observing a heavy output would be estimated using all of the qubits of a large device, but NISQ devices have appreciable error rates, so we begin with small model circuits and progress to larger ones.
* The quantum volume treats the width and depth of a model circuit with equal importance and measures the largest square shaped (i.e., <math>m = d</math>) model circuit a quantum computer can implement successfully on average.
* Given a model circuit <math>U</math>, a circuit-to-circuit transpiler finds an implementation <math>U'</math> for the target system such that <math>1- F_{avg}(U, U') \leq \epsilon \ll 1</math>

==Protocol Description==

'''Function''': ComputeHeavyOutputs<math>(U, m)</math>

'''Input''': <math>U, m</math>

'''Output''': <math>H_U</math>

* Obtain <math>p_U(x)</math> for <math>x \in \{0,1\}^m</math>
* Sort in ascending order <math>p_0 \leq p_1 ... \leq p_{2^m -1}</math>
* <math>p_{med} = (p_{2^{m-1}} + p_{2^{m-1}-1})/2 </math>
* <math>H_U = \{x\in \{0,1\}^m</math> such that <math>p_U(x) > p_{med}\}</math>

'''Function''': ComputeQuantumVolume

'''Output''': Figure of merit: Quantum Volume, <math>V_Q</math>

* For <math>i = 1, 2, ..., m</math>:
** For <math>j = 1, 2, ..., d</math>:
*** <math>d(m) = 0</math>
*** <math>n_h = 0</math>
*** For <math>k = 1, 2, ..., n_c</math>:
**** Pick random model circuit <math>U</math>
**** <math>H_U =</math> ComputeHeavyOutputs<math>(U, m)</math>
**** Compile <math>U'</math>
**** For <math>l = 1, 2, ..., n_s</math>:
***** Get output <math>x</math>
***** If <math>x\in H_U</math> then <math>n_h = n_h + 1</math>
*** If <math>\frac{n_h-2\sqrt{n_h(n_s-n_h/n_c)}}{n_c n_S} > \frac{2}{3}</math>
**** <math>d(m) = </math>max<math>(d(m), d)</math>
**** Store data <math>(m, d(m))</math>
* Calculate <math>V_Q</math> from stored data, where log<math>_2 V_Q</math> = argmax<math>_m</math> min<math>(m, d(m))</math>

==Further Information==

== Related Papers ==
* Andrew W. Cross et al (2011) arXiv:1811.12926v3: Validating quantum computers using randomized model circuits

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Quantum Volume Estimation

2020-03-23T19:48:31Z

Rhea:

[https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.032328 Quantum Volume (QV)] is a single-number metric that can be measured using a concrete protocol on near-term quantum computers of modest size. The QV method quantifies the largest random circuit of equal width and depth that the computer successfully implements. The quantum volume protocol is strongly related to gate error rate and is influenced by underlying qubit connectivity and gate parallelism. This protocol is based on the performance of random model circuits with a fixed but generic form.

'''Tags:''' [[Certification protocol]], [[Quantum Volume estimation protocol]]

==Assumptions==

* The transpiler is free to use all available tricks and hardware resources to implement the model circuit.
* The transpiler should make an honest attempt to implement the model circuit, and not merely choose a relatively simple operation far from the model circuit that nevertheless produces the heavy outputs for it.

==Outline==
The quantum volume protocol is strongly related to gate error rate and is influenced by underlying qubit connectivity and gate parallelism. This protocol is based on the performance of random model circuits with a fixed but generic form.

A model circuit is consists of <math>d</math> layers of random permutations of the <math>m</math> different qubit labels, followed by random two-qubit gates. When the circuit width m is odd, one of the qubits is idle in each layer. Each two-qubit gate used in the previous step is sampled from the [[Haar measure on SU(4)]].

Heavy output generation problem is used to define if the model circuit mentioned above is fully implemented in practice. From the outputs of all the implementations of the model circuit, we get an ideal output distribution. From this, we get the set of output probabilities and we can obtain the median of this set. The heavy outputs are the outputs for which the output probability will be greater than the median of the set of probabilities. The heavy output generation problem is to produce a
set of output strings such that more than two-thirds are
heavy.

To evaluate heavy output generation, we implement model circuits using the gate set provided by the target system, using the available hardware. For this purpose, a quantum circuit-to-circuit transpiler is used, which finds an implementation of the model circuit, where the approximation error.

This method to compute the quantum volume of a device consists of the following steps:

* The Quantum transpiler tries to implement the model circuit such that the approximation error is limited. From here, we get the distribution for the implementation of the model circuit, which we use to calculate the probability of sampling a heavy output.
* The heavy outputs are also computed using the ideal output distribution of the model circuit.
* The probability of observing a heavy output by implementing a randomly selected depth <math>d</math> model circuit is also computed using the probability of sampling a heavy output computed in the step above.
* We define the achievable depth <math>d(m)</math> to be the largest <math>d</math> such that we are confident that the probability of observing a heavy output is greater than <math>2/3</math> (as the heavy output generation problem is to produce a set of output strings such that more than two-thirds are heavy.)
* The data of achievable depth is gathered by sweeping over values of width <math>m</math> and depth <math>d</math> of the model circuit.
* Using all the data gathered, the quantum volume is computed. The quantum volume treats the width and depth of a model circuit with equal importance and measures the largest square shaped (i.e., <math>m = d</math>) model circuit a quantum computer can implement successfully on average.

==Notation==

* <math>U</math>: Model circuit
* <math>U'</math>: Implementation of the model circuit by the quantum transpiler
* <math>m</math>: width of the model circuit
* <math>d</math>: depth of the model circuit
* <math>F_{avg}(U, U')</math>: Average fidelity between <math>U</math> and <math>U'</math>
* <math>\epsilon</math>: approximation error
* <math>d(m)</math>: Achievable depth, which is the largest <math>d</math> such that we are confident that the probability of observing a heavy output is greater than <math>2/3</math>
* <math>V_Q</math>: Quantum Volume
* <math>H_U</math>: Set of heavy outputs for a model circuit <math>U</math>
* <math>x</math>: Outcome of executing <math>U'</math>, which is a observable bit string, <math>x \in \{0,1\}^m</math>
* <math>p_U(x)</math>: Ideal output distribution for <math>U</math>. <math>p_U(x) = |\langle x|U|0\rangle|^2</math>
* <math>p_{med}</math>: median of the set of probabilities
* <math>n_c</math>: Number of repetitions, <math>n_c>100</math>
* <math>n_s</math>: Number of repetitions

==Hardware Requirements==

* Quantum Computing device with a gate set
* Measurement device

==Properties==

* '''Figure of merit''': Quantum Volume
* Quantum computing systems with high-fidelity operations, high connectivity, large calibrated gate sets, and circuit rewriting tool chains are expected to have higher quantum volumes
* The protocol can be implemented with any universal programmable quantum computing device
* The method used to compute the heavy outputs from the ideal output distribution of the model circuit scales exponentially with the width <math>m</math>.
* Ideally, the probability of observing a heavy output would be estimated using all of the qubits of a large device, but NISQ devices have appreciable error rates, so we begin with small model circuits and progress to larger ones.
* The quantum volume treats the width and depth of a model circuit with equal importance and measures the largest square shaped (i.e., <math>m = d</math>) model circuit a quantum computer can implement successfully on average.
* Given a model circuit <math>U</math>, a circuit-to-circuit transpiler finds an implementation <math>U'</math> for the target system such that <math>1- F_{avg}(U, U') \leq \epsilon \ll 1</math>

==Protocol Description==

'''Function''': ComputeHeavyOutputs<math>(U, m)</math>

'''Input''': <math>U, m</math>

'''Output''': <math>H_U</math>

* Obtain <math>p_U(x)</math> for <math>x \in \{0,1\}^m</math>
* Sort in ascending order <math>p_0 \leq p_1 ... \leq p_{2^m -1}</math>
* <math>p_{med} = (p_{2^{m-1}} + p_{2^{m-1}-1})/2 </math>
* <math>H_U = \{x\in \{0,1\}^m</math> such that <math>p_U(x) > p_{med}\}</math>

'''Function''': ComputeQuantumVolume

'''Output''': Figure of merit: Quantum Volume, <math>V_Q</math>

* For <math>i = 1, 2, ..., m</math>:
** For <math>j = 1, 2, ..., d</math>:
*** <math>d(m) = 0</math>
*** <math>n_h = 0</math>
*** For <math>k = 1, 2, ..., n_c</math>:
**** Pick random model circuit <math>U</math>
**** <math>H_U =</math> ComputeHeavyOutputs<math>(U, m)</math>
**** Compile <math>U'</math>
**** For <math>l = 1, 2, ..., n_s</math>:
***** Get output <math>x</math>
***** If <math>x\in H_U</math> then <math>n_h = n_h + 1</math>
*** If <math>\frac{n_h-2\sqrt{n_h(n_s-n_h/n_c)}}{n_c n_S} > \frac{2}{3}</math>
**** <math>d(m) = </math>max<math>(d(m), d)</math>
**** Store data <math>(m, d(m))</math>
* Calculate <math>V_Q</math> from stored data, where log<math>_2 V_Q</math> = argmax<math>_m</math> min<math>(m, d(m))</math>

==Further Information==

== Related Papers ==
* Andrew W. Cross et al (2011) arXiv:1811.12926v3: Validating quantum computers using randomized model circuits

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Quantum Volume Technique

2020-03-23T19:40:58Z

Rhea: Created page with "==Functionality Description== This process is used to estimate the Quantum Volume, which is a single-number metric that can be measured using a concrete protocol on near-term..."

==Functionality Description==
This process is used to estimate the Quantum Volume, which is a single-number metric that can be measured using a concrete protocol on near-term quantum computers of modest size. The QV method quantifies the largest random circuit of equal width and depth that the computer successfully implements. The [[figure of merit]] here is Quantum Volume

==Protocols==
* [[Quantum Volume Estimation]]

==Properties==
* Figure of merit: Quantum Volume
* Quantum computing systems with high-fidelity operations, high connectivity, large calibrated gate sets, and circuit rewriting tool chains are expected to have higher quantum volumes.

==Related Papers==
* Andrew W. Cross et al (2011) arXiv:1811.12926v3: Validating quantum computers using randomized model circuits

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Certification Library

2020-03-23T19:38:33Z

Rhea:

{| class="wikitable"
!width="40%"|Technique
!width="100%"|Protocols
|-
|rowspan="1"|[[Cross Entropy Benchmarking]]||[[Cross Entropy Benchmarking]]
|-
|rowspan="1"|[[Cycle Benchmarking]]||[[Cycle Benchmarking]]
|-
|rowspan="1"|[[Hamiltonian and Phase Estimation]]||[[Hamiltonian and Phase Estimation]]
|-
|rowspan="1"|[[Fidelity Estimation]]||[[Direct Fidelity Estimation]]
|-
|rowspan="1"|[[Fidelity Witnessing]]||[[Fidelity witnesses for fermionic quantum simulations]]
|-
|rowspan="2"|[[Process Tomography]]||[[Full Quantum Process Tomography with Linear inversion]]
|-
|[[Quantum Gate Set Tomography]]
|-
|rowspan="3"|[[Randomised Benchmarking]]||[[Interleaved Randomised Benchmarking]]
|-
|[[Purity Benchmarking]]
|-
|[[Standard Randomised Benchmarking]]
|-
|rowspan="7"|[[State Tomography]]||[[Compressed Sensing Tomography]]
|-
|[[Full Quantum state tomography with Linear Inversion]]
|-
|[[Full Quantum state tomography with Maximum Likelihood Estimation]]
|-
|[[Full Quantum state tomography with Bayesian mean estimation (BME)]]
|-
|[[Full Quantum state tomography using confidence regions]]
|-
|[[Matrix Product State tomography]]
|-
|[[Tensor Network Tomography]]
|-
|rowspan="1"|[[Quantum Volume Technique]]||[[Quantum Volume Estimation]]
|-

Quantum Volume Estimation

2020-03-23T19:35:39Z

Rhea:

[https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.032328 Quantum Volume (QV)] is a single-number metric that can be measured using a concrete protocol on near-term quantum computers of modest size. The QV method quantifies the largest random circuit of equal width and depth that the computer successfully implements. The quantum volume protocol is strongly related to gate error rate and is influenced by underlying qubit connectivity and gate parallelism. This protocol is based on the performance of random model circuits with a fixed but generic form.

'''Tags:''' [[Certification protocol]], [[Quantum Volume estimation protocol]]

==Assumptions==

* The transpiler is free to use all available tricks and hardware resources to implement the model circuit.
* The transpiler should make an honest attempt to implement the model circuit, and not merely choose a relatively simple operation far from the model circuit that nevertheless produces the heavy outputs for it.

==Outline==
The quantum volume protocol is strongly related to gate error rate and is influenced by underlying qubit connectivity and gate parallelism. This protocol is based on the performance of random model circuits with a fixed but generic form.

A model circuit is consists of <math>d</math> layers of random permutations of the <math>m</math> different qubit labels, followed by random two-qubit gates. When the circuit width m is odd, one of the qubits is idle in each layer. Each two-qubit gate used in the previous step is sampled from the [[Haar measure on SU(4)]].

Heavy output generation problem is used to define if the model circuit mentioned above is fully implemented in practice. From the outputs of all the implementations of the model circuit, we get an ideal output distribution. From this, we get the set of output probabilities and we can obtain the median of this set. The heavy outputs are the outputs for which the output probability will be greater than the median of the set of probabilities. The heavy output generation problem is to produce a
set of output strings such that more than two-thirds are
heavy.

To evaluate heavy output generation, we implement model circuits using the gate set provided by the target system, using the available hardware. For this purpose, a quantum circuit-to-circuit transpiler is used, which finds an implementation of the model circuit, where the approximation error.

This method to compute the quantum volume of a device consists of the following steps:

* The Quantum transpiler tries to implement the model circuit such that the approximation error is limited. From here, we get the distribution for the implementation of the model circuit, which we use to calculate the probability of sampling a heavy output.
* The heavy outputs are also computed using the ideal output distribution of the model circuit.
* The probability of observing a heavy output by implementing a randomly selected depth <math>d</math> model circuit is also computed using the probability of sampling a heavy output computed in the step above.
* We define the achievable depth <math>d(m)</math> to be the largest <math>d</math> such that we are confident that the probability of observing a heavy output is greater than <math>2/3</math> (as the heavy output generation problem is to produce a set of output strings such that more than two-thirds are heavy.)
* The data of achievable depth is gathered by sweeping over values of width <math>m</math> and depth <math>d</math> of the model circuit.
* Using all the data gathered, the quantum volume is computed. The quantum volume treats the width and depth of a model circuit with equal importance and measures the largest square shaped (i.e., <math>m = d</math>) model circuit a quantum computer can implement successfully on average.

==Notation==

* <math>U</math>: Model circuit
* <math>U'</math>: Implementation of the model circuit by the quantum transpiler
* <math>m</math>: width of the model circuit
* <math>d</math>: depth of the model circuit
* <math>F_{avg}(U, U')</math>: Average fidelity between <math>U</math> and <math>U'</math>
* <math>\epsilon</math>: approximation error
* <math>d(m)</math>: Achievable depth, which is the largest <math>d</math> such that we are confident that the probability of observing a heavy output is greater than <math>2/3</math>
* <math>V_Q</math>: Quantum Volume
* <math>H_U</math>: Set of heavy outputs for a model circuit <math>U</math>
* <math>x</math>: Outcome of executing <math>U'</math>, which is a observable bit string, <math>x \in \{0,1\}^m</math>
* <math>p_U(x)</math>: Ideal output distribution for <math>U</math>. <math>p_U(x) = |\langle x|U|0\rangle|^2</math>
* <math>p_{med}</math>: median of the set of probabilities
* <math>n_c</math>: Number of repetitions, <math>n_c>100</math>
* <math>n_s</math>: Number of repetitions

==Hardware Requirements==

* Quantum Computing device with a gate set
* Measurement device

==Properties==

* '''Figure of merit''': Quantum Volume
* Quantum computing systems with high-fidelity operations, high connectivity, large calibrated gate sets, and circuit rewriting tool chains are expected to have higher quantum volumes
* The protocol can be implemented with any universal programmable quantum computing device
* The method used to compute the heavy outputs from the ideal output distribution of the model circuit scales exponentially with the width <math>m</math>.
* Ideally, the probability of observing a heavy output would be estimated using all of the qubits of a large device, but NISQ devices have appreciable error rates, so we begin with small model circuits and progress to larger ones.
* The quantum volume treats the width and depth of a model circuit with equal importance and measures the largest square shaped (i.e., <math>m = d</math>) model circuit a quantum computer can implement successfully on average.
* Given a model circuit <math>U</math>, a circuit-to-circuit transpiler finds an implementation <math>U'</math> for the target system such that <math>1- F_{avg}(U, U') \leq \epsilon \ll 1</math>

Quantum Volume Estimation

2020-03-23T15:58:16Z

Rhea: Created page with "[https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.032328 Quantum Volume (QV)] is a single-number metric that can be measured using a concrete protocol on near-term q..."

Quantum Cheque

2020-02-05T18:16:08Z

Rhea: /* Protocol Description */

[https://link.springer.com/article/10.1007/s11128-016-1273-4 Quantum Cheque] is a private-key quantum money scheme that allows trusted banks to issue perfectly secure and verifiable quantum cheque books to account holders and also enables them to undertake monetary transactions with other parties. The quantum cheque is secure against [[non-signalling]] adversaries and cannot be counterfeited.

'''Tags:''' Private-Key Quantum Money, Digital Signature Schemes, [[Quantum Key Distribution]], Quantum One-Way Function.

== Assumptions ==
* The protocol assumes the account holder and the bank, to be honest. The bank is a trusted party, however, the branches may or may not be trusted.
* This protocol assumes perfect state preparations, transmissions, and measurements.
* The protocol takes the assumption that the Digital Signature and the Quantum key distribution schemes are unconditionally secure.
* The digital signature scheme is a black box where any unconditionally secure digital signature scheme can be used which must satisfy the following security conditions of unforgeability and non-repudiation. [[Quantum Digital Signature]] is one of the examples of a digital signature scheme which can be used.

==Outline==

This protocol allows a quantum cheque to be issued using quantum cheque books to the bank customers. The customers can then carry forward transactions in a perfectly secure manner and these cheques can be en-cashed after being verified by the trusted bank or its branches, that communicate with the main branch securely. The quantum cheque follows all the properties of a classical cheque - verifiable by a trusted bank, cannot be disavowed by the issuer and cannot be counterfeited by an adversary. </br>

The entire bank transaction process can be divided into three stages, Gen, where the cheque book is generated for the account holder, Sign, where the account holder prepares a cheque and issues it to the third party and Verify, where the third party en-cashes the cheque depending upon its validity.

* '''Gen'''
** In this stage, the bank first creates a cheque book and a key for the account holder.
** Then, the bank and the account holder create a shared key using [[Quantum Key Distribution]]. Both parties agree upon an unconditionally secure digital signature scheme. The account holder stores his private key safely with himself and shares the public key with the bank.
** The bank then prepares <math>n</math> GHZ triplet states (link to page) and stores only the third entangled qubit of every GHZ in the database, while handing over the first two qubits of every GHZ state to the account holder. Along with this, the bank also creates and shares a corresponding unique serial number for this cheque.
** Finally, the account holder has stored his identity, shared key, private key, public key, serial number and first two entangled qubits of every GHZ triplet state whereas the bank has stored each account holder's identity, shared key, public key, serial number and third entangled qubit of every GHZ triplet states in its database.<br></br>
* '''Sign'''
** In this stage, the issuer's key and amount to be signed is taken as input to produce a quantum state which acts like a cheque.
** Each account holder prepares <math>n</math> quantum states to issue a check worth some amount <math>M</math>. This quantum state is prepared by the quantum one-way function (Link to the page) which takes the shared key, identity of the account holder, a random number and the amount of money <math>M</math> to be signed as the input. Every state from these <math>n</math> quantum states is individually encoded with one of the account holder's entangled qubit of the GHZ states using Quantum Secret Sharing: Splitting of quantum information (Link to the page). The account holder then signs the serial number with their digital signature.
** The quantum cheque thus produces by the account holder contains the information - the identity of the account holder, serial number, generated a random number, digital signature, amount of money signed and the other entangled qubit of the GHZ state.<br></br>
* '''Verify'''
** In this method, the quantum cheque is verified by the trusted bank or its branches and the validity is checked.
** When the quantum cheque is produced at any valid branches of the bank by a third party, the information is securely communicated to the main branch. At the branch, an initial verification is carried by considering the identity of the issuer of the cheque, the serial number of the cheque and the digital signature of the issuer. If the cheque is valid, the verification process is continued, otherwise, the cheque is destroyed and the process is discontinued.
** The main branch performs a measurement on its copy of third entangled qubit of every GHZ triplet state which was stored in the database and securely communicates this result to the branch.
** The branch recovers the quantum state that was prepared by the account holder to issue the cheque, using the information received from the main branch. The branch also computes this same quantum states using the stored account holder's information like the shared key, the identity of the account holder, a random number and the amount of money <math>M</math> as input for verification. A swap test(Link to the page) is performed on both these states and if the cheque is only accepted if this test passes, else it is destroyed and aborted.

This scheme is proven to be impossible to counterfeit and impossible to repudiate. The quantum cheque is a quantum state and thus it cannot be copied or stolen by any eavesdropper, ensuring that only one copy of the quantum cheque exists.

==Requirements==
* Secure quantum channel to connect the branches of the bank to the main branch.
* Secure quantum channel to connect the bank to the account holder and to connect any other third party to the bank.
* This protocol required quantum memory to store the issuer's quantum state of the cheque if the protocol is not running in real-time.
* Private database for both account holder and bank.
* Measurement devices for the account holder and the main branch of the bank.

==Knowledge Graph==

{{graph}}
==Properties==

* In the signing process, the quantum one-way function used to create the cheque for the account holder is assumed to take polynomial time to compute and is hard to invert.
* In the verification process, the bank sets a thresholding security parameter <math>\kappa</math>. The swap test is passed if <math>\{ \langle\psi^{(i)}|\psi^{,(i)}\rangle \geq \kappa\}_{i=1:l}</math>
* No quantum memory would be required for the account holder to store the quantum check if the transaction is occurring in real-time.
* This protocol can be realized, efficiently, with few qubit systems, without compromising on the security
* Security: This protocol is impossible to counterfeit and non-repudiation by signatory is impossible here.

==Notation==
* <math>k</math>: Shared key between account holder and bank where <math>k \in \{0,1\}^L</math>.
* <math>\Pi</math>: A decided Digital Signature scheme, which could also be [[Quantum Digital Signature]] or any other classical digital signature scheme, where <math>\Pi=(Gen, Sign, Vrfy)</math>. Here <math>Gen</math> is the digital signature generation funciton, <math>Sign</math> is the signing function and <math>Vrfy</math> is the function which verifies if the digital signature belongs to a party.
* <math>id</math>: Account holder's identity.
* <math>pk</math>: Account holder's public key.
* <math>sk</math>: Account holder's private key.
* <math>s</math>: Unique serial number where <math>s \in \{0,1\}^{l(n)}</math>.
* <math>{|\phi^{(i)}\rangle_{GHZ}}</math> or <math>{\rho^{(i)}_{GHZ}}</math>: <math>n</math> GHZ triplet states where,
<div style="text-align: center;"><math>|\phi^{(i)}\rangle_{GHZ} = \rho^{(i)}_{GHZ} = \frac{1}{\sqrt{2}}(|0^{(i)}\rangle_{A_1}|0^{(i)}\rangle_{A_2}|0^{(i)}\rangle_{B} + |1^{(i)}\rangle_{A_1}|1^{(i)}\rangle_{A_2}|1^{(i)}\rangle_{B})</math></div>
* <math>{\rho^{(i)}_{A_1}}</math>: First entangled qubit from every GHZ triplet state which is given to account holder.
* <math>{\rho^{(i)}_{A_2}}</math>: Second entangled qubit from every GHZ triplet state which is given to account holder.
* <math>{\rho^{(i)}_{B}}</math>: First entangled qubit from every GHZ triplet state which stays with bank.
* <math>r</math>: Generated random number where <math>r \in \{0,1\}^L</math>.
* <math>M</math>: Amount of money account holder signs on the cheque.
* <math>f</math>: Quantum one way function.
* <math>x\mid\mid</math>y: concatenation of two bit strings <math>x</math> and <math>y</math>.
* <math>{|\psi^{(i)}\rangle}</math>: quantum state prepared by the account where,
<div style="text-align: center;"><math>|\psi^{(i)}\rangle = f(k \mid\mid id\mid\mid r\mid\mid M\mid\mid i)</math></div>. This quantum state is also denoted as <math>\alpha_i|0\rangle + \beta_i|1\rangle</math>.
* <math>|\Phi^\pm\rangle</math>: Bell state <math>\frac{|00\rangle \pm |11\rangle}{\sqrt{2}}</math>
* <math>|\Psi^\pm\rangle</math>: Bell state <math>\frac{|10\rangle \pm |01\rangle}{\sqrt{2}}</math>
* <math>\sigma</math>: In <math>\chi</math>, output of signing with <math>s</math>.
* <math>\chi</math>: Quantum cheque where,
<div style="text-align: center;"><math>
\chi = (id, s, r, \sigma, M, \{{\rho^{(i)}_{A_1}}\}_{i=1:n})
</math></div>
* <math>\kappa</math>: Threshold constant set as a security parameter by the bank in the swap test.
* <math>|\psi^{,(i)}\rangle</math>: Quantum state prepared by the bank for verification.

==Protocol Description==

'''Stage 1''': Gen</br>
'''Output''': Account holder now holds <math>(id, pk, sk, k, s, \{\rho^{(i)}_{A_1}\rho^{(i)}_{A_2}\}_{i=1:n})</math> and Bank now holds <math>(id, pk, k, s, \{\rho^{(i)}_{B}\}_{i=1:n})</math> in their private databases.

* Account holder and Bank create <math>k</math>.
* Account holder and Bank agree on <math>\Pi</math>.
* Account holder submits <math>pk</math> to Bank.
* Account holder secretly stores <math>sk</math>.
* For <math>i = 1, 2, ...n</math>:
** Bank generates GHZ triple state <math>|\phi^{(i)}\rangle_{GHZ}</math>
** Bank stores <math>\rho^{(i)}_{B}</math> in their private database.
** Bank gives <math>\rho^{(i)}_{A_1}\rho^{(i)}_{A_2}</math> to account holder.
* Bank prepares <math>s</math> and shares it with the account holder.
* For <math>i = 1, 2, ...n</math>:
** Account holder stores <math>\rho^{(i)}_{A_1}\rho^{(i)}_{A_2}</math> privately.
* Account holder now holds <math>(id, pk, sk, k, s, \{\rho^{(i)}_{A_1}\rho^{(i)}_{A_2}\}_{i=1:n})</math>
* Bank now holds <math>(id, pk, k, s, \{\rho^{(i)}_{B}\}_{i=1:n})</math>

'''Stage 2''': Sign </br>
'''Output''': Account holder produces <math>\chi</math>

* Account holder generates <math>r</math>.
* For <math>i = 1, 2, ...n</math>:
** Account holder prepares <math>|\psi^{(i)}\rangle = f(k \mid\mid id\mid\mid r\mid\mid M\mid\mid i)</math>.
** Account holder encodes <math>|\psi^{(i)}\rangle</math> with <math>|\phi^{(i)}\rangle_{GHZ}</math> by combining <math>|\psi^{(i)}\rangle</math> with <math>\rho^{(i)}_{A_1}</math> and performing a bell measurement on the two.
<div style="text-align: center;"><math>
|\phi^{(i)}\rangle = |\psi^{(i)}\rangle \otimes |\phi^{(i)}\rangle_{GHZ}
</math></div>
** Based on the measurement, account holder performs the suitable error correction, by applying the corresponding Pauli matrix, on <math>\rho^{(i)}_{A_1}</math>:
<div style="text-align: center;"><math>
|\Phi^{(+)}\rangle \xrightarrow{} I,
|\Phi^{(-)}\rangle \xrightarrow{} \sigma_z,
|\Psi^{(+)}\rangle \xrightarrow{} \sigma_x,
|\Phi^{(-)}\rangle \xrightarrow{} \sigma_y,
</math></div>
* Account holder signs the serial number <math>s</math> using <math>\Pi</math>, as <math>\sigma \xleftarrow{} Sign_{sk}(s)</math>.
* Account holder produces <math>\chi</math>.

'''Stage 3''': Verify </br>
'''Output''': Cheque gets accepted or the process is aborted and the cheque is destroyed.

* Third party produces the <math>\chi</math> at a valid branch of the bank.
* Branch communicates with main branch of the bank and checks validity of <math>(id, s)</math> and runs <math>{Vrfy}_{pk}(\sigma, s)</math>.
** If invalid:
*** Bank aborts the process and destroys the cheque.
** else:
*** Bank continues the verification process.
* Main branch of the bank performs the measurement in Hadamard basis on its copy of <math>\rho^{(i)}_{B}</math> and obtains outcome <math>|+\rangle</math> or <math>|-\rangle</math>.
* Main branch communicates this result with the local branch.
* For <math>i = 1, 2, ...n</math>:
** Based on outcome, branch performs the corresponding Pauli matrix operation on <math>\rho^{(i)}_{A_2}</math> to recover <math>|\Psi^{(i)}\rangle</math> : <math>|+\rangle \xrightarrow{} I, |-\rangle \xrightarrow{} \sigma_z</math>
* For <math>i = 1, 2, ...n</math>:
** Bank computes <math>|\Psi^{,(i)}\rangle </math>, where,
<div style="text-align: center;"><math>
|\Psi^{,(i)}\rangle = f(k \mid\mid id\mid\mid r\mid\mid M\mid\mid i)
</math></div>
* For <math>i = 1, 2, ...n</math>:
** Bank performs swap test on <math>|\Psi^{(i)}\rangle</math> and <math>|\Psi^{,(i)}\rangle</math>.
*** If <math>\langle\Psi^{(i)}|\Psi^{,(i)}\rangle \geq \kappa</math>:
**** Bank aborts the process and destroys the cheque.
*** else:
**** Bank continues the verification process.
* Bank accepts the cheque.

==Further information==

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Quantum Cheque

2020-02-05T18:14:38Z

Rhea:

[https://link.springer.com/article/10.1007/s11128-016-1273-4 Quantum Cheque] is a private-key quantum money scheme that allows trusted banks to issue perfectly secure and verifiable quantum cheque books to account holders and also enables them to undertake monetary transactions with other parties. The quantum cheque is secure against [[non-signalling]] adversaries and cannot be counterfeited.

'''Tags:''' Private-Key Quantum Money, Digital Signature Schemes, [[Quantum Key Distribution]], Quantum One-Way Function.

== Assumptions ==
* The protocol assumes the account holder and the bank, to be honest. The bank is a trusted party, however, the branches may or may not be trusted.
* This protocol assumes perfect state preparations, transmissions, and measurements.
* The protocol takes the assumption that the Digital Signature and the Quantum key distribution schemes are unconditionally secure.
* The digital signature scheme is a black box where any unconditionally secure digital signature scheme can be used which must satisfy the following security conditions of unforgeability and non-repudiation. [[Quantum Digital Signature]] is one of the examples of a digital signature scheme which can be used.

==Outline==

This protocol allows a quantum cheque to be issued using quantum cheque books to the bank customers. The customers can then carry forward transactions in a perfectly secure manner and these cheques can be en-cashed after being verified by the trusted bank or its branches, that communicate with the main branch securely. The quantum cheque follows all the properties of a classical cheque - verifiable by a trusted bank, cannot be disavowed by the issuer and cannot be counterfeited by an adversary. </br>

The entire bank transaction process can be divided into three stages, Gen, where the cheque book is generated for the account holder, Sign, where the account holder prepares a cheque and issues it to the third party and Verify, where the third party en-cashes the cheque depending upon its validity.

* '''Gen'''
** In this stage, the bank first creates a cheque book and a key for the account holder.
** Then, the bank and the account holder create a shared key using [[Quantum Key Distribution]]. Both parties agree upon an unconditionally secure digital signature scheme. The account holder stores his private key safely with himself and shares the public key with the bank.
** The bank then prepares <math>n</math> GHZ triplet states (link to page) and stores only the third entangled qubit of every GHZ in the database, while handing over the first two qubits of every GHZ state to the account holder. Along with this, the bank also creates and shares a corresponding unique serial number for this cheque.
** Finally, the account holder has stored his identity, shared key, private key, public key, serial number and first two entangled qubits of every GHZ triplet state whereas the bank has stored each account holder's identity, shared key, public key, serial number and third entangled qubit of every GHZ triplet states in its database.<br></br>
* '''Sign'''
** In this stage, the issuer's key and amount to be signed is taken as input to produce a quantum state which acts like a cheque.
** Each account holder prepares <math>n</math> quantum states to issue a check worth some amount <math>M</math>. This quantum state is prepared by the quantum one-way function (Link to the page) which takes the shared key, identity of the account holder, a random number and the amount of money <math>M</math> to be signed as the input. Every state from these <math>n</math> quantum states is individually encoded with one of the account holder's entangled qubit of the GHZ states using Quantum Secret Sharing: Splitting of quantum information (Link to the page). The account holder then signs the serial number with their digital signature.
** The quantum cheque thus produces by the account holder contains the information - the identity of the account holder, serial number, generated a random number, digital signature, amount of money signed and the other entangled qubit of the GHZ state.<br></br>
* '''Verify'''
** In this method, the quantum cheque is verified by the trusted bank or its branches and the validity is checked.
** When the quantum cheque is produced at any valid branches of the bank by a third party, the information is securely communicated to the main branch. At the branch, an initial verification is carried by considering the identity of the issuer of the cheque, the serial number of the cheque and the digital signature of the issuer. If the cheque is valid, the verification process is continued, otherwise, the cheque is destroyed and the process is discontinued.
** The main branch performs a measurement on its copy of third entangled qubit of every GHZ triplet state which was stored in the database and securely communicates this result to the branch.
** The branch recovers the quantum state that was prepared by the account holder to issue the cheque, using the information received from the main branch. The branch also computes this same quantum states using the stored account holder's information like the shared key, the identity of the account holder, a random number and the amount of money <math>M</math> as input for verification. A swap test(Link to the page) is performed on both these states and if the cheque is only accepted if this test passes, else it is destroyed and aborted.

This scheme is proven to be impossible to counterfeit and impossible to repudiate. The quantum cheque is a quantum state and thus it cannot be copied or stolen by any eavesdropper, ensuring that only one copy of the quantum cheque exists.

==Requirements==
* Secure quantum channel to connect the branches of the bank to the main branch.
* Secure quantum channel to connect the bank to the account holder and to connect any other third party to the bank.
* This protocol required quantum memory to store the issuer's quantum state of the cheque if the protocol is not running in real-time.
* Private database for both account holder and bank.
* Measurement devices for the account holder and the main branch of the bank.

==Knowledge Graph==

{{graph}}
==Properties==

* In the signing process, the quantum one-way function used to create the cheque for the account holder is assumed to take polynomial time to compute and is hard to invert.
* In the verification process, the bank sets a thresholding security parameter <math>\kappa</math>. The swap test is passed if <math>\{ \langle\psi^{(i)}|\psi^{,(i)}\rangle \geq \kappa\}_{i=1:l}</math>
* No quantum memory would be required for the account holder to store the quantum check if the transaction is occurring in real-time.
* This protocol can be realized, efficiently, with few qubit systems, without compromising on the security
* Security: This protocol is impossible to counterfeit and non-repudiation by signatory is impossible here.

==Notation==
* <math>k</math>: Shared key between account holder and bank where <math>k \in \{0,1\}^L</math>.
* <math>\Pi</math>: A decided Digital Signature scheme, which could also be [[Quantum Digital Signature]] or any other classical digital signature scheme, where <math>\Pi=(Gen, Sign, Vrfy)</math>. Here <math>Gen</math> is the digital signature generation funciton, <math>Sign</math> is the signing function and <math>Vrfy</math> is the function which verifies if the digital signature belongs to a party.
* <math>id</math>: Account holder's identity.
* <math>pk</math>: Account holder's public key.
* <math>sk</math>: Account holder's private key.
* <math>s</math>: Unique serial number where <math>s \in \{0,1\}^{l(n)}</math>.
* <math>{|\phi^{(i)}\rangle_{GHZ}}</math> or <math>{\rho^{(i)}_{GHZ}}</math>: <math>n</math> GHZ triplet states where,
<div style="text-align: center;"><math>|\phi^{(i)}\rangle_{GHZ} = \rho^{(i)}_{GHZ} = \frac{1}{\sqrt{2}}(|0^{(i)}\rangle_{A_1}|0^{(i)}\rangle_{A_2}|0^{(i)}\rangle_{B} + |1^{(i)}\rangle_{A_1}|1^{(i)}\rangle_{A_2}|1^{(i)}\rangle_{B})</math></div>
* <math>{\rho^{(i)}_{A_1}}</math>: First entangled qubit from every GHZ triplet state which is given to account holder.
* <math>{\rho^{(i)}_{A_2}}</math>: Second entangled qubit from every GHZ triplet state which is given to account holder.
* <math>{\rho^{(i)}_{B}}</math>: First entangled qubit from every GHZ triplet state which stays with bank.
* <math>r</math>: Generated random number where <math>r \in \{0,1\}^L</math>.
* <math>M</math>: Amount of money account holder signs on the cheque.
* <math>f</math>: Quantum one way function.
* <math>x\mid\mid</math>y: concatenation of two bit strings <math>x</math> and <math>y</math>.
* <math>{|\psi^{(i)}\rangle}</math>: quantum state prepared by the account where,
<div style="text-align: center;"><math>|\psi^{(i)}\rangle = f(k \mid\mid id\mid\mid r\mid\mid M\mid\mid i)</math></div>. This quantum state is also denoted as <math>\alpha_i|0\rangle + \beta_i|1\rangle</math>.
* <math>|\Phi^\pm\rangle</math>: Bell state <math>\frac{|00\rangle \pm |11\rangle}{\sqrt{2}}</math>
* <math>|\Psi^\pm\rangle</math>: Bell state <math>\frac{|10\rangle \pm |01\rangle}{\sqrt{2}}</math>
* <math>\sigma</math>: In <math>\chi</math>, output of signing with <math>s</math>.
* <math>\chi</math>: Quantum cheque where,
<div style="text-align: center;"><math>
\chi = (id, s, r, \sigma, M, \{{\rho^{(i)}_{A_1}}\}_{i=1:n})
</math></div>
* <math>\kappa</math>: Threshold constant set as a security parameter by the bank in the swap test.
* <math>|\psi^{,(i)}\rangle</math>: Quantum state prepared by the bank for verification.

==Protocol Description==

'''Stage 1''': Gen</br>
'''Output''': Account holder now holds <math>(id, pk, sk, k, s, \{\rho^{(i)}_{A_1}\rho^{(i)}_{A_2}\}_{i=1:n})</math> and Bank now holds <math>(id, pk, k, s, \{\rho^{(i)}_{B}\}_{i=1:n})</math> in their private databases.

* Account holder and Bank create <math>k</math>.
* Account holder and Bank agree on <math>\Pi</math>.
* Account holder submits <math>pk</math> to Bank.
* Account holder secretly stores <math>sk</math>.
* For <math>i = 1, 2, ...n</math>:
** Bank generates GHZ triple state <math>|\phi^{(i)}\rangle_{GHZ}</math>
** Bank stores <math>\rho^{(i)}_{B}</math> in their private database.
** Bank gives <math>\rho^{(i)}_{A_1}\rho^{(i)}_{A_2}</math> to account holder.
* Bank prepares <math>s</math> and shares it with the account holder.
* For <math>i = 1, 2, ...n</math>:
** Account holder stores <math>\rho^{(i)}_{A_1}\rho^{(i)}_{A_2}</math> privately.
* Account holder now holds <math>(id, pk, sk, k, s, \{\rho^{(i)}_{A_1}\rho^{(i)}_{A_2}\}_{i=1:n})</math>
* Bank now holds <math>(id, pk, k, s, {\rho^{(i)}_{B}\}_{i=1:n})}</math>

'''Stage 2''': Sign </br>
'''Output''': Account holder produces <math>\chi</math>

* Account holder generates <math>r</math>.
* For <math>i = 1, 2, ...n</math>:
** Account holder prepares <math>|\psi^{(i)}\rangle = f(k \mid\mid id\mid\mid r\mid\mid M\mid\mid i)</math>.
** Account holder encodes <math>|\psi^{(i)}\rangle</math> with <math>|\phi^{(i)}\rangle_{GHZ}</math> by combining <math>|\psi^{(i)}\rangle</math> with <math>\rho^{(i)}_{A_1}</math> and performing a bell measurement on the two.
<div style="text-align: center;"><math>
|\phi^{(i)}\rangle = |\psi^{(i)}\rangle \otimes |\phi^{(i)}\rangle_{GHZ}
</math></div>
** Based on the measurement, account holder performs the suitable error correction, by applying the corresponding Pauli matrix, on <math>\rho^{(i)}_{A_1}</math>:
<div style="text-align: center;"><math>
|\Phi^{(+)}\rangle \xrightarrow{} I,
|\Phi^{(-)}\rangle \xrightarrow{} \sigma_z,
|\Psi^{(+)}\rangle \xrightarrow{} \sigma_x,
|\Phi^{(-)}\rangle \xrightarrow{} \sigma_y,
</math></div>
* Account holder signs the serial number <math>s</math> using <math>\Pi</math>, as <math>\sigma \xleftarrow{} Sign_{sk}(s)</math>.
* Account holder produces <math>\chi</math>.

'''Stage 3''': Verify </br>
'''Output''': Cheque gets accepted or the process is aborted and the cheque is destroyed.

* Third party produces the <math>\chi</math> at a valid branch of the bank.
* Branch communicates with main branch of the bank and checks validity of <math>(id, s)</math> and runs <math>{Vrfy}_{pk}(\sigma, s)</math>.
** If invalid:
*** Bank aborts the process and destroys the cheque.
** else:
*** Bank continues the verification process.
* Main branch of the bank performs the measurement in Hadamard basis on its copy of <math>\rho^{(i)}_{B}</math> and obtains outcome <math>|+\rangle</math> or <math>|-\rangle</math>.
* Main branch communicates this result with the local branch.
* For <math>i = 1, 2, ...n</math>:
** Based on outcome, branch performs the corresponding Pauli matrix operation on <math>\rho^{(i)}_{A_2}</math> to recover <math>|\Psi^{(i)}\rangle</math> : <math>|+\rangle \xrightarrow{} I, |-\rangle \xrightarrow{} \sigma_z</math>
* For <math>i = 1, 2, ...n</math>:
** Bank computes <math>|\Psi^{,(i)}\rangle </math>, where,
<div style="text-align: center;"><math>
|\Psi^{,(i)}\rangle = f(k \mid\mid id\mid\mid r\mid\mid M\mid\mid i)
</math></div>
* For <math>i = 1, 2, ...n</math>:
** Bank performs swap test on <math>|\Psi^{(i)}\rangle</math> and <math>|\Psi^{,(i)}\rangle</math>.
*** If <math>\langle\Psi^{(i)}|\Psi^{,(i)}\rangle \geq \kappa</math>:
**** Bank aborts the process and destroys the cheque.
*** else:
**** Bank continues the verification process.
* Bank accepts the cheque.

==Further information==

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Quantum Cheque

2020-02-05T17:56:28Z

Rhea:

[https://link.springer.com/article/10.1007/s11128-016-1273-4 Quantum Cheque] is a private-key quantum money scheme that allows trusted banks to issue perfectly secure and verifiable quantum cheque books to account holders and also enables them to undertake monetary transactions with other parties. The quantum cheque is secure against [[non-signalling]] adversaries and cannot be counterfeited.

'''Tags:''' Private-Key Quantum Money, Digital Signature Schemes, [[Quantum Key Distribution]], Quantum One-Way Function.

== Assumptions ==
* The protocol assumes the account holder and the bank, to be honest. The bank is a trusted party, however, the branches may or may not be trusted.
* This protocol assumes perfect state preparations, transmissions, and measurements.
* The protocol takes the assumption that the Digital Signature and the Quantum key distribution schemes are unconditionally secure.
* The digital signature scheme is a black box where any unconditionally secure digital signature scheme can be used which must satisfy the following security conditions of unforgeability and non-repudiation. [[Quantum Digital Signature]] is one of the examples of a digital signature scheme which can be used.

==Outline==

This protocol allows a quantum cheque to be issued using quantum cheque books to the bank customers. The customers can then carry forward transactions in a perfectly secure manner and these cheques can be en-cashed after being verified by the trusted bank or its branches, that communicate with the main branch securely. The quantum cheque follows all the properties of a classical cheque - verifiable by a trusted bank, cannot be disavowed by the issuer and cannot be counterfeited by an adversary. </br>

The entire bank transaction process can be divided into three stages, Gen, where the cheque book is generated for the account holder, Sign, where the account holder prepares a cheque and issues it to the third party and Verify, where the third party en-cashes the cheque depending upon its validity.

* '''Gen'''
** In this stage, the bank first creates a cheque book and a key for the account holder.
** Then, the bank and the account holder create a shared key using [[Quantum Key Distribution]]. Both parties agree upon an unconditionally secure digital signature scheme. The account holder stores his private key safely with himself and shares the public key with the bank.
** The bank then prepares <math>n</math> GHZ triplet states (link to page) and stores only the third entangled qubit of every GHZ in the database, while handing over the first two qubits of every GHZ state to the account holder. Along with this, the bank also creates and shares a corresponding unique serial number for this cheque.
** Finally, the account holder has stored his identity, shared key, private key, public key, serial number and first two entangled qubits of every GHZ triplet state whereas the bank has stored each account holder's identity, shared key, public key, serial number and third entangled qubit of every GHZ triplet states in its database.<br></br>
* '''Sign'''
** In this stage, the issuer's key and amount to be signed is taken as input to produce a quantum state which acts like a cheque.
** Each account holder prepares <math>n</math> quantum states to issue a check worth some amount <math>M</math>. This quantum state is prepared by the quantum one-way function (Link to the page) which takes the shared key, identity of the account holder, a random number and the amount of money <math>M</math> to be signed as the input. Every state from these <math>n</math> quantum states is individually encoded with one of the account holder's entangled qubit of the GHZ states using Quantum Secret Sharing: Splitting of quantum information (Link to the page). The account holder then signs the serial number with their digital signature.
** The quantum cheque thus produces by the account holder contains the information - the identity of the account holder, serial number, generated a random number, digital signature, amount of money signed and the other entangled qubit of the GHZ state.<br></br>
* '''Verify'''
** In this method, the quantum cheque is verified by the trusted bank or its branches and the validity is checked.
** When the quantum cheque is produced at any valid branches of the bank by a third party, the information is securely communicated to the main branch. At the branch, an initial verification is carried by considering the identity of the issuer of the cheque, the serial number of the cheque and the digital signature of the issuer. If the cheque is valid, the verification process is continued, otherwise, the cheque is destroyed and the process is discontinued.
** The main branch performs a measurement on its copy of third entangled qubit of every GHZ triplet state which was stored in the database and securely communicates this result to the branch.
** The branch recovers the quantum state that was prepared by the account holder to issue the cheque, using the information received from the main branch. The branch also computes this same quantum states using the stored account holder's information like the shared key, the identity of the account holder, a random number and the amount of money <math>M</math> as input for verification. A swap test(Link to the page) is performed on both these states and if the cheque is only accepted if this test passes, else it is destroyed and aborted.

This scheme is proven to be impossible to counterfeit and impossible to repudiate. The quantum cheque is a quantum state and thus it cannot be copied or stolen by any eavesdropper, ensuring that only one copy of the quantum cheque exists.

==Requirements==
* Secure quantum channel to connect the branches of the bank to the main branch.
* Secure quantum channel to connect the bank to the account holder and to connect any other third party to the bank.
* This protocol required quantum memory to store the issuer's quantum state of the cheque if the protocol is not running in real-time.
* Private database for both account holder and bank.
* Measurement devices for the account holder and the main branch of the bank.

==Knowledge Graph==

{{graph}}
==Properties==

* In the signing process, the quantum one-way function used to create the cheque for the account holder is assumed to take polynomial time to compute and is hard to invert.
* In the verification process, the bank sets a thresholding security parameter <math>\kappa</math>. The swap test is passed if <math>\{ \langle\psi^{(i)}|\psi^{,(i)}\rangle \geq \kappa\}_{i=1:l}</math>
* No quantum memory would be required for the account holder to store the quantum check if the transaction is occurring in real-time.
* This protocol can be realized, efficiently, with few qubit systems, without compromising on the security
* Security: This protocol is impossible to counterfeit and non-repudiation by signatory is impossible here.

==Notation==
* <math>k</math>: Shared key between account holder and bank where <math>k \in \{0,1\}^L</math>.
* <math>\Pi</math>: A decided Digital Signature scheme, which could also be [[Quantum Digital Signature]] or any other classical digital signature scheme, where <math>\Pi=(Gen, Sign, Vrfy)</math>. Here <math>Gen</math> is the digital signature generation funciton, <math>Sign</math> is the signing function and <math>Vrfy</math> is the function which verifies if the digital signature belongs to a party.
* <math>id</math>: Account holder's identity.
* <math>pk</math>: Account holder's public key.
* <math>sk</math>: Account holder's private key.
* <math>s</math>: Unique serial number where <math>s \in \{0,1\}^{l(n)}</math>.
* <math>{|\phi^{(i)}\rangle_{GHZ}}</math> or <math>{\rho^{(i)}_{GHZ}}</math>: <math>n</math> GHZ triplet states where,
<div style="text-align: center;"><math>|\phi^{(i)}\rangle_{GHZ} = \rho^{(i)}_{GHZ} = \frac{1}{\sqrt{2}}(|0^{(i)}\rangle_{A_1}|0^{(i)}\rangle_{A_2}|0^{(i)}\rangle_{B} + |1^{(i)}\rangle_{A_1}|1^{(i)}\rangle_{A_2}|1^{(i)}\rangle_{B})</math></div>
* <math>{|\phi^{(i)}\rangle_{A_1}}</math>: First entangled qubit from every GHZ triplet state which is given to account holder.
* <math>{|\phi^{(i)}\rangle_{A_2}}</math>: Second entangled qubit from every GHZ triplet state which is given to account holder.
* <math>{|\phi^{(i)}\rangle_{B}}</math>: First entangled qubit from every GHZ triplet state which stays with bank.
* <math>r</math>: Generated random number where <math>r \in \{0,1\}^L</math>.
* <math>M</math>: Amount of money account holder signs on the cheque.
* <math>f</math>: Quantum one way function.
* <math>x\mid\mid</math>y: concatenation of two bit strings <math>x</math> and <math>y</math>.
* <math>{|\psi^{(i)}\rangle}</math>: quantum state prepared by the account where,
<div style="text-align: center;"><math>|\psi^{(i)}\rangle = f(k \mid\mid id\mid\mid r\mid\mid M\mid\mid i)</math></div>. This quantum state is also denoted as <math>\alpha_i|0\rangle + \beta_i|1\rangle</math>.
* <math>|\Phi^\pm\rangle</math>: Bell state <math>\frac{|00\rangle \pm |11\rangle}{\sqrt{2}}</math>
* <math>|\Psi^\pm\rangle</math>: Bell state <math>\frac{|10\rangle \pm |01\rangle}{\sqrt{2}}</math>
* <math>\sigma</math>: In <math>\chi</math>, output of signing with <math>s</math>.
* <math>\chi</math>: Quantum cheque where,
<div style="text-align: center;"><math>
\chi = (id, s, r, \sigma, M, \{{|\phi^{(i)}\rangle_{A_1}}\}_{i=1:n})
</math></div>
* <math>\kappa</math>: Threshold constant set as a security parameter by the bank in the swap test.
* <math>|\psi^{,(i)}\rangle</math>: Quantum state prepared by the bank for verification.

==Protocol Description==

'''Stage 1''': Gen</br>
'''Output''': Account holder now holds <math>(id, pk, sk, k, s, \{|\phi^{(i)}\rangle_{A_1}|\phi^{(i)}\rangle_{A_2}\}_{i=1:n})</math> and Bank now holds <math>(id, pk, k, s, {\|\phi^{(i)}\rangle_{B}\}_{i=1:n})}</math> in their private databases.

* Account holder and Bank create <math>k</math>.
* Account holder and Bank agree on <math>\Pi</math>.
* Account holder submits <math>pk</math> to Bank.
* Account holder secretly stores <math>sk</math>.
* For <math>i = 1, 2, ...n</math>:
** Bank generates GHZ triple state <math>|\phi^{(i)}\rangle_{GHZ}</math>
** Bank stores <math>|\phi^{(i)}\rangle_{B}</math> in their private database.
** Bank gives <math>|\phi^{(i)}\rangle_{A_1}|\phi^{(i)}\rangle_{A_2}</math> to account holder.
* Bank prepares <math>s</math> and shares it with the account holder.
* For <math>i = 1, 2, ...n</math>:
** Account holder stores <math>|\phi^{(i)}\rangle_{A_1}|\phi^{(i)}\rangle_{A_2}</math> privately.
* Account holder now holds <math>(id, pk, sk, k, s, \{|\phi^{(i)}\rangle_{A_1}|\phi^{(i)}\rangle_{A_2}\}_{i=1:n})</math>
* Bank now holds <math>(id, pk, k, s, {|\phi^{(i)}\rangle_{B}\}_{i=1:n})}</math>

'''Stage 2''': Sign </br>
'''Output''': Account holder produces <math>\chi</math>

* Account holder generates <math>r</math>.
* For <math>i = 1, 2, ...n</math>:
** Account holder prepares <math>|\psi^{(i)}\rangle = f(k \mid\mid id\mid\mid r\mid\mid M\mid\mid i)</math>.
** Account holder encodes <math>|\psi^{(i)}\rangle</math> with <math>|\phi^{(i)}\rangle_{GHZ}</math> by combining <math>|\psi^{(i)}\rangle</math> with <math>|\phi^{(i)}\rangle_{A_1}</math> and performing a bell measurement on the two.
<div style="text-align: center;"><math>
|\phi^{(i)}\rangle = |\psi^{(i)}\rangle \otimes |\phi^{(i)}\rangle_{GHZ}
</math></div>
** Based on the measurement, account holder performs the suitable error correction, by applying the corresponding Pauli matrix, on <math>|\phi^{(i)}\rangle_{A_1}</math>:
<div style="text-align: center;"><math>
|\Phi^{(+)}\rangle \xrightarrow{} I,
|\Phi^{(-)}\rangle \xrightarrow{} \sigma_z,
|\Psi^{(+)}\rangle \xrightarrow{} \sigma_x,
|\Phi^{(-)}\rangle \xrightarrow{} \sigma_y,
</math></div>
* Account holder signs the serial number <math>s</math> using <math>\Pi</math>, as <math>\sigma \xleftarrow{} Sign_{sk}(s)</math>.
* Account holder produces <math>\chi</math>.

'''Stage 3''': Verify </br>
'''Output''': Cheque gets accepted or the process is aborted and the cheque is destroyed.

* Third party produces the <math>\chi</math> at a valid branch of the bank.
* Branch communicates with main branch of the bank and checks validity of <math>(id, s)</math> and runs <math>{Vrfy}_{pk}(\sigma, s)</math>.
** If invalid:
*** Bank aborts the process and destroys the cheque.
** else:
*** Bank continues the verification process.
* Main branch of the bank performs the measurement in Hadamard basis on its copy of <math>|\phi^{(i)}\rangle_{B}</math> and obtains outcome <math>|+\rangle</math> or <math>|-\rangle</math>.
* Main branch communicates this result with the local branch.
* For <math>i = 1, 2, ...n</math>:
** Based on outcome, branch performs the corresponding Pauli matrix operation on <math>|\phi^{(i)}\rangle_{A_2}</math> to recover <math>|\Psi^{(i)}\rangle</math> : <math>|+\rangle \xrightarrow{} I, |-\rangle \xrightarrow{} \sigma_z</math>
* For <math>i = 1, 2, ...n</math>:
** Bank computes <math>|\Psi^{,(i)}\rangle </math>, where,
<div style="text-align: center;"><math>
|\Psi^{,(i)}\rangle = f(k \mid\mid id\mid\mid r\mid\mid M\mid\mid i)
</math></div>
* For <math>i = 1, 2, ...n</math>:
** Bank performs swap test on <math>|\Psi^{(i)}\rangle</math> and <math>|\Psi^{,(i)}\rangle</math>.
*** If <math>\langle\Psi^{(i)}|\Psi^{,(i)}\rangle \geq \kappa</math>:
**** Bank aborts the process and destroys the cheque.
*** else:
**** Bank continues the verification process.
* Bank accepts the cheque.

==Further information==

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Quantum Cheque

2020-02-04T13:41:26Z

Rhea: /* Notation */

[https://link.springer.com/article/10.1007/s11128-016-1273-4 Quantum Cheque] is a private-key quantum money scheme that allows trusted banks to issue perfectly secure and verifiable quantum cheque books to account holders and also enables them to undertake monetary transactions with other parties. The quantum cheque is secure against [[non-signalling]] adversaries and cannot be counterfeited.

'''Tags:''' Private-Key Quantum Money, Digital Signature Schemes, [[Quantum Key Distribution]], Quantum One-Way Function.

== Assumptions ==
* The protocol assumes the account holder and the bank, to be honest. The bank is a trusted party, however, the branches may or may not be trusted.
* This protocol assumes perfect state preparations, transmissions, and measurements.
* The protocol takes the assumption that the Digital Signature and the Quantum key distribution schemes are unconditionally secure.
* The digital signature scheme is a black box where any unconditionally secure digital signature scheme can be used which must satisfy the following security conditions of unforgeability and non-repudiation. [[Quantum Digital Signature]] is one of the examples of a digital signature scheme which can be used.

==Outline==

This protocol allows a quantum cheque to be issued using quantum cheque books to the bank customers. The customers can then carry forward transactions in a perfectly secure manner and these cheques can be en-cashed after being verified by the trusted bank or its branches, that communicate with the main branch securely. The quantum cheque follows all the properties of a classical cheque - verifiable by a trusted bank, cannot be disavowed by the issuer and cannot be counterfeited by an adversary. </br>

The entire bank transaction process can be divided into three stages, Gen, where the cheque book is generated for the account holder, Sign, where the account holder prepares a cheque and issues it to the third party and Verify, where the third party en-cashes the cheque depending upon its validity.

* '''Gen'''
** In this stage, the bank first creates a cheque book and a key for the account holder.
** Then, the bank and the account holder create a shared key using [[Quantum Key Distribution]]. Both parties agree upon an unconditionally secure digital signature scheme. The account holder stores his private key safely with himself and shares the public key with the bank.
** The bank then prepares <math>n</math> GHZ triplet states (link to page) and stores only the third entangled qubit of every GHZ in the database, while handing over the first two qubits of every GHZ state to the account holder. Along with this, the bank also creates and shares a corresponding unique serial number for this cheque.
** Finally, the account holder has stored his identity, shared key, private key, public key, serial number and first two entangled qubits of every GHZ triplet state whereas the bank has stored each account holder's identity, shared key, public key, serial number and third entangled qubit of every GHZ triplet states in its database.<br></br>
* '''Sign'''
** In this stage, the issuer's key and amount to be signed is taken as input to produce a quantum state which acts like a cheque.
** Each account holder prepares <math>n</math> quantum states to issue a check worth some amount <math>M</math>. This quantum state is prepared by the quantum one-way function (Link to the page) which takes the shared key, identity of the account holder, a random number and the amount of money <math>M</math> to be signed as the input. Every state from these <math>n</math> quantum states is individually encoded with one of the account holder's entangled qubit of the GHZ states using Quantum Secret Sharing: Splitting of quantum information (Link to the page). The account holder then signs the serial number with their digital signature.
** The quantum cheque thus produces by the account holder contains the information - the identity of the account holder, serial number, generated a random number, digital signature, amount of money signed and the other entangled qubit of the GHZ state.<br></br>
* '''Verify'''
** In this method, the quantum cheque is verified by the trusted bank or its branches and the validity is checked.
** When the quantum cheque is produced at any valid branches of the bank by a third party, the information is securely communicated to the main branch. At the branch, an initial verification is carried by considering the identity of the issuer of the cheque, the serial number of the cheque and the digital signature of the issuer. If the cheque is valid, the verification process is continued, otherwise, the cheque is destroyed and the process is discontinued.
** The main branch performs a measurement on its copy of third entangled qubit of every GHZ triplet state which was stored in the database and securely communicates this result to the branch.
** The branch recovers the quantum state that was prepared by the account holder to issue the cheque, using the information received from the main branch. The branch also computes this same quantum states using the stored account holder's information like the shared key, the identity of the account holder, a random number and the amount of money <math>M</math> as input for verification. A swap test(Link to the page) is performed on both these states and if the cheque is only accepted if this test passes, else it is destroyed and aborted.

This scheme is proven to be impossible to counterfeit and impossible to repudiate. The quantum cheque is a quantum state and thus it cannot be copied or stolen by any eavesdropper, ensuring that only one copy of the quantum cheque exists.

==Requirements==
* Secure quantum channel to connect the branches of the bank to the main branch.
* Secure quantum channel to connect the bank to the account holder and to connect any other third party to the bank.
* This protocol required quantum memory to store the issuer's quantum state of the cheque if the protocol is not running in real-time.
* Private database for both account holder and bank.
* Measurement devices for the account holder and the main branch of the bank.

==Knowledge Graph==

{{graph}}
==Properties==

* In the signing process, the quantum one-way function used to create the cheque for the account holder is assumed to take polynomial time to compute and is hard to invert.
* In the verification process, the bank sets a thresholding security parameter <math>\kappa</math>. The swap test is passed if <math>\{ \langle\psi^{(i)}|\psi^{,(i)}\rangle \geq \kappa\}_{i=1:l}</math>
* No quantum memory would be required for the account holder to store the quantum check if the transaction is occurring in real-time.
* This protocol can be realized, efficiently, with few qubit systems, without compromising on the security
* Security: This protocol is impossible to counterfeit and non-repudiation by signatory is impossible here.

==Notation==
* <math>k</math>: Shared key between account holder and bank where <math>k \in \{0,1\}^L</math>.
* <math>\Pi</math>: A decided Digital Signature scheme, which could also be [[Quantum Digital Signature]] or any other classical digital signature scheme, where <math>\Pi=(Gen, Sign, Vrfy)</math>. Here <math>Gen</math> is the digital signature generation funciton, <math>Sign</math> is the signing function and <math>Vrfy</math> is the function which verifies if the digital signature belongs to a party.
* <math>id</math>: Account holder's identity.
* <math>pk</math>: Account holder's public key.
* <math>sk</math>: Account holder's private key.
* <math>s</math>: Unique serial number where <math>s \in \{0,1\}^{l(n)}</math>.
* <math>{|\phi^{(i)}\rangle_{GHZ}}</math>: <math>n</math> GHZ triplet states where,
<div style="text-align: center;"><math>|\phi^{(i)}\rangle_{GHZ} = \frac{1}{\sqrt{2}}(|0^{(i)}\rangle_{A_1}|0^{(i)}\rangle_{A_2}|0^{(i)}\rangle_{B} + |1^{(i)}\rangle_{A_1}|1^{(i)}\rangle_{A_2}|1^{(i)}\rangle_{B})</math></div>
* <math>{|\phi^{(i)}\rangle_{A_1}}</math>: First entangled qubit from every GHZ triplet state which is given to account holder.
* <math>{|\phi^{(i)}\rangle_{A_2}}</math>: Second entangled qubit from every GHZ triplet state which is given to account holder.
* <math>{|\phi^{(i)}\rangle_{B}}</math>: First entangled qubit from every GHZ triplet state which stays with bank.
* <math>r</math>: Generated random number where <math>r \in \{0,1\}^L</math>.
* <math>M</math>: Amount of money account holder signs on the cheque.
* <math>f</math>: Quantum one way function.
* <math>x\mid\mid</math>y: concatenation of two bit strings <math>x</math> and <math>y</math>.
* <math>{|\psi^{(i)}\rangle}</math>: quantum state prepared by the account where,
<div style="text-align: center;"><math>|\psi^{(i)}\rangle = f(k \mid\mid id\mid\mid r\mid\mid M\mid\mid i)</math></div>. This quantum state is also denoted as <math>\alpha_i|0\rangle + \beta_i|1\rangle</math>.
* <math>|\Phi^\pm\rangle</math>: Bell state <math>\frac{|00\rangle \pm |11\rangle}{\sqrt{2}}</math>
* <math>|\Psi^\pm\rangle</math>: Bell state <math>\frac{|10\rangle \pm |01\rangle}{\sqrt{2}}</math>
* <math>\sigma</math>: In <math>\chi</math>, output of signing with <math>s</math>.
* <math>\chi</math>: Quantum cheque where,
<div style="text-align: center;"><math>
\chi = (id, s, r, \sigma, M, \{{|\phi^{(i)}\rangle_{A_1}}\}_{i=1:n})
</math></div>
* <math>\kappa</math>: Threshold constant set as a security parameter by the bank in the swap test.
* <math>|\psi^{,(i)}\rangle</math>: Quantum state prepared by the bank for verification.

==Protocol Description==

'''Stage 1''': Gen</br>
'''Output''': Account holder now holds <math>(id, pk, sk, k, s, \{|\phi^{(i)}\rangle_{A_1}|\phi^{(i)}\rangle_{A_2}\}_{i=1:n})</math> and Bank now holds <math>(id, pk, k, s, {\|\phi^{(i)}\rangle_{B}\}_{i=1:n})}</math> in their private databases.

* Account holder and Bank create <math>k</math>.
* Account holder and Bank agree on <math>\Pi</math>.
* Account holder submits <math>pk</math> to Bank.
* Account holder secretly stores <math>sk</math>.
* For <math>i = 1, 2, ...n</math>:
** Bank generates GHZ triple state <math>|\phi^{(i)}\rangle_{GHZ}</math>
** Bank stores <math>|\phi^{(i)}\rangle_{B}</math> in their private database.
** Bank gives <math>|\phi^{(i)}\rangle_{A_1}|\phi^{(i)}\rangle_{A_2}</math> to account holder.
* Bank prepares <math>s</math> and shares it with the account holder.
* For <math>i = 1, 2, ...n</math>:
** Account holder stores <math>|\phi^{(i)}\rangle_{A_1}|\phi^{(i)}\rangle_{A_2}</math> privately.
* Account holder now holds <math>(id, pk, sk, k, s, \{|\phi^{(i)}\rangle_{A_1}|\phi^{(i)}\rangle_{A_2}\}_{i=1:n})</math>
* Bank now holds <math>(id, pk, k, s, {|\phi^{(i)}\rangle_{B}\}_{i=1:n})}</math>

'''Stage 2''': Sign </br>
'''Output''': Account holder produces <math>\chi</math>

* Account holder generates <math>r</math>.
* For <math>i = 1, 2, ...n</math>:
** Account holder prepares <math>|\psi^{(i)}\rangle = f(k \mid\mid id\mid\mid r\mid\mid M\mid\mid i)</math>.
** Account holder encodes <math>|\psi^{(i)}\rangle</math> with <math>|\phi^{(i)}\rangle_{GHZ}</math> by combining <math>|\psi^{(i)}\rangle</math> with <math>|\phi^{(i)}\rangle_{A_1}</math> and performing a bell measurement on the two.
<div style="text-align: center;"><math>
|\phi^{(i)}\rangle = |\psi^{(i)}\rangle \otimes |\phi^{(i)}\rangle_{GHZ}
</math></div>
** Based on the measurement, account holder performs the suitable error correction, by applying the corresponding Pauli matrix, on <math>|\phi^{(i)}\rangle_{A_1}</math>:
<div style="text-align: center;"><math>
|\Phi^{(+)}\rangle \xrightarrow{} I,
|\Phi^{(-)}\rangle \xrightarrow{} \sigma_z,
|\Psi^{(+)}\rangle \xrightarrow{} \sigma_x,
|\Phi^{(-)}\rangle \xrightarrow{} \sigma_y,
</math></div>
* Account holder signs the serial number <math>s</math> using <math>\Pi</math>, as <math>\sigma \xleftarrow{} Sign_{sk}(s)</math>.
* Account holder produces <math>\chi</math>.

'''Stage 3''': Verify </br>
'''Output''': Cheque gets accepted or the process is aborted and the cheque is destroyed.

* Third party produces the <math>\chi</math> at a valid branch of the bank.
* Branch communicates with main branch of the bank and checks validity of <math>(id, s)</math> and runs <math>{Vrfy}_{pk}(\sigma, s)</math>.
** If invalid:
*** Bank aborts the process and destroys the cheque.
** else:
*** Bank continues the verification process.
* Main branch of the bank performs the measurement in Hadamard basis on its copy of <math>|\phi^{(i)}\rangle_{B}</math> and obtains outcome <math>|+\rangle</math> or <math>|-\rangle</math>.
* Main branch communicates this result with the local branch.
* For <math>i = 1, 2, ...n</math>:
** Based on outcome, branch performs the corresponding Pauli matrix operation on <math>|\phi^{(i)}\rangle_{A_2}</math> to recover <math>|\Psi^{(i)}\rangle</math> : <math>|+\rangle \xrightarrow{} I, |-\rangle \xrightarrow{} \sigma_z</math>
* For <math>i = 1, 2, ...n</math>:
** Bank computes <math>|\Psi^{,(i)}\rangle </math>, where,
<div style="text-align: center;"><math>
|\Psi^{,(i)}\rangle = f(k \mid\mid id\mid\mid r\mid\mid M\mid\mid i)
</math></div>
* For <math>i = 1, 2, ...n</math>:
** Bank performs swap test on <math>|\Psi^{(i)}\rangle</math> and <math>|\Psi^{,(i)}\rangle</math>.
*** If <math>\langle\Psi^{(i)}|\Psi^{,(i)}\rangle \geq \kappa</math>:
**** Bank aborts the process and destroys the cheque.
*** else:
**** Bank continues the verification process.
* Bank accepts the cheque.

==Further information==

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Quantum Cheque

2020-02-04T13:34:20Z

Rhea:

[https://link.springer.com/article/10.1007/s11128-016-1273-4 Quantum Cheque] is a private-key quantum money scheme that allows trusted banks to issue perfectly secure and verifiable quantum cheque books to account holders and also enables them to undertake monetary transactions with other parties. The quantum cheque is secure against [[non-signalling]] adversaries and cannot be counterfeited.

'''Tags:''' Private-Key Quantum Money, Digital Signature Schemes, [[Quantum Key Distribution]], Quantum One-Way Function.

== Assumptions ==
* The protocol assumes the account holder and the bank, to be honest. The bank is a trusted party, however, the branches may or may not be trusted.
* This protocol assumes perfect state preparations, transmissions, and measurements.
* The protocol takes the assumption that the Digital Signature and the Quantum key distribution schemes are unconditionally secure.
* The digital signature scheme is a black box where any unconditionally secure digital signature scheme can be used which must satisfy the following security conditions of unforgeability and non-repudiation. [[Quantum Digital Signature]] is one of the examples of a digital signature scheme which can be used.

==Outline==

This protocol allows a quantum cheque to be issued using quantum cheque books to the bank customers. The customers can then carry forward transactions in a perfectly secure manner and these cheques can be en-cashed after being verified by the trusted bank or its branches, that communicate with the main branch securely. The quantum cheque follows all the properties of a classical cheque - verifiable by a trusted bank, cannot be disavowed by the issuer and cannot be counterfeited by an adversary. </br>

The entire bank transaction process can be divided into three stages, Gen, where the cheque book is generated for the account holder, Sign, where the account holder prepares a cheque and issues it to the third party and Verify, where the third party en-cashes the cheque depending upon its validity.

* '''Gen'''
** In this stage, the bank first creates a cheque book and a key for the account holder.
** Then, the bank and the account holder create a shared key using [[Quantum Key Distribution]]. Both parties agree upon an unconditionally secure digital signature scheme. The account holder stores his private key safely with himself and shares the public key with the bank.
** The bank then prepares <math>n</math> GHZ triplet states (link to page) and stores only the third entangled qubit of every GHZ in the database, while handing over the first two qubits of every GHZ state to the account holder. Along with this, the bank also creates and shares a corresponding unique serial number for this cheque.
** Finally, the account holder has stored his identity, shared key, private key, public key, serial number and first two entangled qubits of every GHZ triplet state whereas the bank has stored each account holder's identity, shared key, public key, serial number and third entangled qubit of every GHZ triplet states in its database.<br></br>
* '''Sign'''
** In this stage, the issuer's key and amount to be signed is taken as input to produce a quantum state which acts like a cheque.
** Each account holder prepares <math>n</math> quantum states to issue a check worth some amount <math>M</math>. This quantum state is prepared by the quantum one-way function (Link to the page) which takes the shared key, identity of the account holder, a random number and the amount of money <math>M</math> to be signed as the input. Every state from these <math>n</math> quantum states is individually encoded with one of the account holder's entangled qubit of the GHZ states using Quantum Secret Sharing: Splitting of quantum information (Link to the page). The account holder then signs the serial number with their digital signature.
** The quantum cheque thus produces by the account holder contains the information - the identity of the account holder, serial number, generated a random number, digital signature, amount of money signed and the other entangled qubit of the GHZ state.<br></br>
* '''Verify'''
** In this method, the quantum cheque is verified by the trusted bank or its branches and the validity is checked.
** When the quantum cheque is produced at any valid branches of the bank by a third party, the information is securely communicated to the main branch. At the branch, an initial verification is carried by considering the identity of the issuer of the cheque, the serial number of the cheque and the digital signature of the issuer. If the cheque is valid, the verification process is continued, otherwise, the cheque is destroyed and the process is discontinued.
** The main branch performs a measurement on its copy of third entangled qubit of every GHZ triplet state which was stored in the database and securely communicates this result to the branch.
** The branch recovers the quantum state that was prepared by the account holder to issue the cheque, using the information received from the main branch. The branch also computes this same quantum states using the stored account holder's information like the shared key, the identity of the account holder, a random number and the amount of money <math>M</math> as input for verification. A swap test(Link to the page) is performed on both these states and if the cheque is only accepted if this test passes, else it is destroyed and aborted.

This scheme is proven to be impossible to counterfeit and impossible to repudiate. The quantum cheque is a quantum state and thus it cannot be copied or stolen by any eavesdropper, ensuring that only one copy of the quantum cheque exists.

==Requirements==
* Secure quantum channel to connect the branches of the bank to the main branch.
* Secure quantum channel to connect the bank to the account holder and to connect any other third party to the bank.
* This protocol required quantum memory to store the issuer's quantum state of the cheque if the protocol is not running in real-time.
* Private database for both account holder and bank.
* Measurement devices for the account holder and the main branch of the bank.

==Knowledge Graph==

{{graph}}
==Properties==

* In the signing process, the quantum one-way function used to create the cheque for the account holder is assumed to take polynomial time to compute and is hard to invert.
* In the verification process, the bank sets a thresholding security parameter <math>\kappa</math>. The swap test is passed if <math>\{ \langle\psi^{(i)}|\psi^{,(i)}\rangle \geq \kappa\}_{i=1:l}</math>
* No quantum memory would be required for the account holder to store the quantum check if the transaction is occurring in real-time.
* This protocol can be realized, efficiently, with few qubit systems, without compromising on the security
* Security: This protocol is impossible to counterfeit and non-repudiation by signatory is impossible here.

==Notation==
* <math>k</math>: Shared key between account holder and bank where <math>k \in \{0,1\}^L</math>.
* <math>\Pi</math>: A decided Digital Signature scheme, which could also be [[Quantum Digital Signature]] or any other classical digital signature scheme, where <math>\Pi=(Gen, Sign, Vrfy)</math>.
* <math>id</math>: Account holder's identity.
* <math>pk</math>: Account holder's public key.
* <math>sk</math>: Account holder's private key.
* <math>s</math>: Unique serial number where <math>s \in \{0,1\}^{l(n)}</math>.
* <math>{|\phi^{(i)}\rangle_{GHZ}}</math>: <math>n</math> GHZ triplet states where,
<div style="text-align: center;"><math>|\phi^{(i)}\rangle_{GHZ} = \frac{1}{\sqrt{2}}(|0^{(i)}\rangle_{A_1}|0^{(i)}\rangle_{A_2}|0^{(i)}\rangle_{B} + |1^{(i)}\rangle_{A_1}|1^{(i)}\rangle_{A_2}|1^{(i)}\rangle_{B})</math></div>
* <math>{|\phi^{(i)}\rangle_{A_1}}</math>: First entangled qubit from every GHZ triplet state which is given to account holder.
* <math>{|\phi^{(i)}\rangle_{A_2}}</math>: Second entangled qubit from every GHZ triplet state which is given to account holder.
* <math>{|\phi^{(i)}\rangle_{B}}</math>: First entangled qubit from every GHZ triplet state which stays with bank.
* <math>r</math>: Generated random number where <math>r \in \{0,1\}^L</math>.
* <math>M</math>: Amount of money account holder signs on the cheque.
* <math>f</math>: Quantum one way function.
* <math>x\mid\mid</math>y: concatenation of two bit strings <math>x</math> and <math>y</math>.
* <math>{|\psi^{(i)}\rangle}</math>: quantum state prepared by the account where,
<div style="text-align: center;"><math>|\psi^{(i)}\rangle = f(k \mid\mid id\mid\mid r\mid\mid M\mid\mid i)</math></div>. This quantum state is also denoted as <math>\alpha_i|0\rangle + \beta_i|1\rangle</math>.
* <math>|\Phi^\pm\rangle</math>: Bell state <math>\frac{|00\rangle \pm |11\rangle}{\sqrt{2}}</math>
* <math>|\Psi^\pm\rangle</math>: Bell state <math>\frac{|10\rangle \pm |01\rangle}{\sqrt{2}}</math>
* <math>\sigma</math>: In <math>\chi</math>, output of signing with <math>s</math>.
* <math>\chi</math>: Quantum cheque where,
<div style="text-align: center;"><math>
\chi = (id, s, r, \sigma, M, \{{|\phi^{(i)}\rangle_{A_1}}\}_{i=1:n})
</math></div>
* <math>\kappa</math>: Threshold constant set as a security parameter by the bank in the swap test.
* <math>|\psi^{,(i)}\rangle</math>: Quantum state prepared by the bank for verification.

==Protocol Description==

'''Stage 1''': Gen</br>
'''Output''': Account holder now holds <math>(id, pk, sk, k, s, \{|\phi^{(i)}\rangle_{A_1}|\phi^{(i)}\rangle_{A_2}\}_{i=1:n})</math> and Bank now holds <math>(id, pk, k, s, {\|\phi^{(i)}\rangle_{B}\}_{i=1:n})}</math> in their private databases.

* Account holder and Bank create <math>k</math>.
* Account holder and Bank agree on <math>\Pi</math>.
* Account holder submits <math>pk</math> to Bank.
* Account holder secretly stores <math>sk</math>.
* For <math>i = 1, 2, ...n</math>:
** Bank generates GHZ triple state <math>|\phi^{(i)}\rangle_{GHZ}</math>
** Bank stores <math>|\phi^{(i)}\rangle_{B}</math> in their private database.
** Bank gives <math>|\phi^{(i)}\rangle_{A_1}|\phi^{(i)}\rangle_{A_2}</math> to account holder.
* Bank prepares <math>s</math> and shares it with the account holder.
* For <math>i = 1, 2, ...n</math>:
** Account holder stores <math>|\phi^{(i)}\rangle_{A_1}|\phi^{(i)}\rangle_{A_2}</math> privately.
* Account holder now holds <math>(id, pk, sk, k, s, \{|\phi^{(i)}\rangle_{A_1}|\phi^{(i)}\rangle_{A_2}\}_{i=1:n})</math>
* Bank now holds <math>(id, pk, k, s, {|\phi^{(i)}\rangle_{B}\}_{i=1:n})}</math>

'''Stage 2''': Sign </br>
'''Output''': Account holder produces <math>\chi</math>

* Account holder generates <math>r</math>.
* For <math>i = 1, 2, ...n</math>:
** Account holder prepares <math>|\psi^{(i)}\rangle = f(k \mid\mid id\mid\mid r\mid\mid M\mid\mid i)</math>.
** Account holder encodes <math>|\psi^{(i)}\rangle</math> with <math>|\phi^{(i)}\rangle_{GHZ}</math> by combining <math>|\psi^{(i)}\rangle</math> with <math>|\phi^{(i)}\rangle_{A_1}</math> and performing a bell measurement on the two.
<div style="text-align: center;"><math>
|\phi^{(i)}\rangle = |\psi^{(i)}\rangle \otimes |\phi^{(i)}\rangle_{GHZ}
</math></div>
** Based on the measurement, account holder performs the suitable error correction, by applying the corresponding Pauli matrix, on <math>|\phi^{(i)}\rangle_{A_1}</math>:
<div style="text-align: center;"><math>
|\Phi^{(+)}\rangle \xrightarrow{} I,
|\Phi^{(-)}\rangle \xrightarrow{} \sigma_z,
|\Psi^{(+)}\rangle \xrightarrow{} \sigma_x,
|\Phi^{(-)}\rangle \xrightarrow{} \sigma_y,
</math></div>
* Account holder signs the serial number <math>s</math> using <math>\Pi</math>, as <math>\sigma \xleftarrow{} Sign_{sk}(s)</math>.
* Account holder produces <math>\chi</math>.

'''Stage 3''': Verify </br>
'''Output''': Cheque gets accepted or the process is aborted and the cheque is destroyed.

* Third party produces the <math>\chi</math> at a valid branch of the bank.
* Branch communicates with main branch of the bank and checks validity of <math>(id, s)</math> and runs <math>{Vrfy}_{pk}(\sigma, s)</math>.
** If invalid:
*** Bank aborts the process and destroys the cheque.
** else:
*** Bank continues the verification process.
* Main branch of the bank performs the measurement in Hadamard basis on its copy of <math>|\phi^{(i)}\rangle_{B}</math> and obtains outcome <math>|+\rangle</math> or <math>|-\rangle</math>.
* Main branch communicates this result with the local branch.
* For <math>i = 1, 2, ...n</math>:
** Based on outcome, branch performs the corresponding Pauli matrix operation on <math>|\phi^{(i)}\rangle_{A_2}</math> to recover <math>|\Psi^{(i)}\rangle</math> : <math>|+\rangle \xrightarrow{} I, |-\rangle \xrightarrow{} \sigma_z</math>
* For <math>i = 1, 2, ...n</math>:
** Bank computes <math>|\Psi^{,(i)}\rangle </math>, where,
<div style="text-align: center;"><math>
|\Psi^{,(i)}\rangle = f(k \mid\mid id\mid\mid r\mid\mid M\mid\mid i)
</math></div>
* For <math>i = 1, 2, ...n</math>:
** Bank performs swap test on <math>|\Psi^{(i)}\rangle</math> and <math>|\Psi^{,(i)}\rangle</math>.
*** If <math>\langle\Psi^{(i)}|\Psi^{,(i)}\rangle \geq \kappa</math>:
**** Bank aborts the process and destroys the cheque.
*** else:
**** Bank continues the verification process.
* Bank accepts the cheque.

==Further information==

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Quantum Cheque

2020-02-04T13:14:04Z

Rhea: /* Assumptions */

[https://link.springer.com/article/10.1007/s11128-016-1273-4 Quantum Cheque] is a private-key quantum money scheme that allows trusted banks to issue perfectly secure and verifiable quantum cheque books to account holders and also enables them to undertake monetary transactions with other parties. The quantum cheque is secure against [[non-signalling]] adversaries and cannot be counterfeited.

'''Tags:''' Private-Key Quantum Money, [[Quantum Digital Signature]], [[Quantum Key Distribution]], Quantum One-Way Function.

== Assumptions ==
* The protocol assumes the account holder and the bank, to be honest. The bank is a trusted party, however, the branches may or may not be trusted.
* This protocol assumes perfect state preparations, transmissions, and measurements.
* The protocol takes the assumption that the Digital Signature and the Quantum key distribution schemes are unconditionally secure.
* The digital signature scheme is a black box where any unconditionally secure digital signature scheme can be used which must satisfy the following security conditions of unforgeability and non-repudiation. [[Quantum Digital Signature]] is one of the examples of a digital signature scheme which can be used.

==Outline==

This protocol allows a quantum cheque to be issued using quantum cheque books to the bank customers. The customers can then carry forward transactions in a perfectly secure manner and these cheques can be en-cashed after being verified by the trusted bank or its branches, that communicate with the main branch securely. The quantum cheque follows all the properties of a classical cheque - verifiable by a trusted bank, cannot be disavowed by the issuer and cannot be counterfeited by an adversary. </br>

The entire bank transaction process can be divided into three stages, Gen, where the cheque book is generated for the account holder, Sign, where the account holder prepares a cheque and issues it to the third party and Verify, where the third party en-cashes the cheque depending upon its validity.

* '''Gen'''
** In this stage, the bank first creates a cheque book and a key for the account holder.
** Then, the bank and the account holder create a shared key using [[Quantum Key Distribution]]. Both parties agree upon the [[Quantum Digital Signature]]. The account holder stores his private key safely with himself and shares the public key with the bank.
** The bank then prepares <math>n</math> GHZ triplet states (link to page) and stores only the third entangled qubit of every GHZ in the database, while handing over the first two qubits of every GHZ state to the account holder. Along with this, the bank also creates and shares a corresponding unique serial number for this cheque.
** Finally, the account holder has stored his identity, shared key, private key, public key, serial number and first two entangled qubits of every GHZ triplet state whereas the bank has stored each account holder's identity, shared key, public key, serial number and third entangled qubit of every GHZ triplet states in its database.<br></br>
* '''Sign'''
** In this stage, the issuer's key and amount to be signed is taken as input to produce a quantum state which acts like a cheque.
** Each account holder prepares <math>n</math> quantum states to issue a check worth some amount <math>M</math>. This quantum state is prepared by the quantum one-way function (Link to the page) which takes the shared key, identity of the account holder, a random number and the amount of money <math>M</math> to be signed as the input. Every state from these <math>n</math> quantum states is individually encoded with one of the account holder's entangled qubit of the GHZ states using Quantum Secret Sharing: Splitting of quantum information (Link to the page). The account holder then signs the serial number with their digital signature.
** The quantum cheque thus produces by the account holder contains the information - the identity of the account holder, serial number, generated a random number, digital signature, amount of money signed and the other entangled qubit of the GHZ state.<br></br>
* '''Verify'''
** In this method, the quantum cheque is verified by the trusted bank or its branches and the validity is checked.
** When the quantum cheque is produced at any valid branches of the bank by a third party, the information is securely communicated to the main branch. At the branch, an initial verification is carried by considering the identity of the issuer of the cheque, the serial number of the cheque and the digital signature of the issuer. If the cheque is valid, the verification process is continued, otherwise, the cheque is destroyed and the process is discontinued.
** The main branch performs a measurement on its copy of third entangled qubit of every GHZ triplet state which was stored in the database and securely communicates this result to the branch.
** The branch recovers the quantum state that was prepared by the account holder to issue the cheque, using the information received from the main branch. The branch also computes this same quantum states using the stored account holder's information like the shared key, the identity of the account holder, a random number and the amount of money <math>M</math> as input for verification. A swap test(Link to the page) is performed on both these states and if the cheque is only accepted if this test passes, else it is destroyed and aborted.

This scheme is proven to be impossible to counterfeit and impossible to repudiate. The quantum cheque is a quantum state and thus it cannot be copied or stolen by any eavesdropper, ensuring that only one copy of the quantum cheque exists.

==Requirements==
* Secure quantum channel to connect the branches of the bank to the main branch.
* Secure quantum channel to connect the bank to the account holder and to connect any other third party to the bank.
* This protocol required quantum memory to store issuer's quantum state of the cheque if the protocol is not running in real time.
* Private database for both account holder and bank.
* Measurement devices for the account holder and the main branch of the bank.

==Knowledge Graph==

{{graph}}
==Properties==

* In the signing process, the quantum one-way function used to create the cheque for the account holder is assumed to take polynomial time to compute and is hard to invert.
* In the verification process, the bank sets a thresholding security parameter <math>\kappa</math>. The swap test is passed if <math>\{ \langle\psi^{(i)}|\psi^{,(i)}\rangle \geq \kappa\}_{i=1:l}</math>
* No quantum memory would be required for the account holder to store the quantum check if the transaction is occurring in real time.
* This protocol can be realized, efficiently, with few qubit systems, without compromising on the security
* Security: This protocol is impossible to counterfeit and non-repudiation by signatory is impossible here.

==Notation==
* <math>k</math>: Shared key between account holder and bank where <math>k \in \{0,1\}^L</math>.
* <math>\Pi</math>: [[Quantum Digital Signature]] scheme, where <math>\Pi=(Gen, Sign, Vrfy)</math>.
* <math>id</math>: Account holder's identity.
* <math>pk</math>: Account holder's public key.
* <math>sk</math>: Account holder's private key.
* <math>s</math>: Unique serial number where <math>s \in \{0,1\}^{l(n)}</math>.
* <math>{|\phi^{(i)}\rangle_{GHZ}}</math>: <math>n</math> GHZ triplet states where,
<div style="text-align: center;"><math>|\phi^{(i)}\rangle_{GHZ} = \frac{1}{\sqrt{2}}(|0^{(i)}\rangle_{A_1}|0^{(i)}\rangle_{A_2}|0^{(i)}\rangle_{B} + |1^{(i)}\rangle_{A_1}|1^{(i)}\rangle_{A_2}|1^{(i)}\rangle_{B})</math></div>
* <math>{|\phi^{(i)}\rangle_{A_1}}</math>: First entangled qubit from every GHZ triplet state which is given to account holder.
* <math>{|\phi^{(i)}\rangle_{A_2}}</math>: Second entangled qubit from every GHZ triplet state which is given to account holder.
* <math>{|\phi^{(i)}\rangle_{B}}</math>: First entangled qubit from every GHZ triplet state which stays with bank.
* <math>r</math>: Generated random number where <math>r \in \{0,1\}^L</math>.
* <math>M</math>: Amount of money account holder signs on the cheque.
* <math>f</math>: Quantum one way function.
* <math>x\mid\mid</math>y: concatenation of two bit strings <math>x</math> and <math>y</math>.
* <math>{|\psi^{(i)}\rangle}</math>: quantum state prepared by the account where,
<div style="text-align: center;"><math>|\psi^{(i)}\rangle = f(k \mid\mid id\mid\mid r\mid\mid M\mid\mid i)</math></div>. This quantum state is also denoted as <math>\alpha_i|0\rangle + \beta_i|1\rangle</math>.
* <math>|\Phi^\pm\rangle</math>: Bell state <math>\frac{|00\rangle \pm |11\rangle}{\sqrt{2}}</math>
* <math>|\Psi^\pm\rangle</math>: Bell state <math>\frac{|10\rangle \pm |01\rangle}{\sqrt{2}}</math>
* <math>\sigma</math>: In <math>\chi</math>, output of signing with <math>s</math>.
* <math>\chi</math>: Quantum cheque where,
<div style="text-align: center;"><math>
\chi = (id, s, r, \sigma, M, \{{|\phi^{(i)}\rangle_{A_1}}\}_{i=1:n})
</math></div>
* <math>\kappa</math>: Threshold constant set as a security parameter by the bank in the swap test.
* <math>|\psi^{,(i)}\rangle</math>: Quantum state prepared by the bank for verification.

==Protocol Description==

'''Stage 1''': Gen</br>
'''Output''': Account holder now holds <math>(id, pk, sk, k, s, \{|\phi^{(i)}\rangle_{A_1}|\phi^{(i)}\rangle_{A_2}\}_{i=1:n})</math> and Bank now holds <math>(id, pk, k, s, {\|\phi^{(i)}\rangle_{B}\}_{i=1:n})}</math> in their private databases.

* Account holder and Bank create <math>k</math>.
* Account holder and Bank agree on <math>\Pi</math>.
* Account holder submits <math>pk</math> to Bank.
* Account holder secretly stores <math>sk</math>.
* For <math>i = 1, 2, ...n</math>:
** Bank generates GHZ triple state <math>|\phi^{(i)}\rangle_{GHZ}</math>
** Bank stores <math>|\phi^{(i)}\rangle_{B}</math> in their private database.
** Bank gives <math>|\phi^{(i)}\rangle_{A_1}|\phi^{(i)}\rangle_{A_2}</math> to account holder.
* Bank prepares <math>s</math> and shares it with the account holder.
* For <math>i = 1, 2, ...n</math>:
** Account holder stores <math>|\phi^{(i)}\rangle_{A_1}|\phi^{(i)}\rangle_{A_2}</math> privately.
* Account holder now holds <math>(id, pk, sk, k, s, \{|\phi^{(i)}\rangle_{A_1}|\phi^{(i)}\rangle_{A_2}\}_{i=1:n})</math>
* Bank now holds <math>(id, pk, k, s, {|\phi^{(i)}\rangle_{B}\}_{i=1:n})}</math>

'''Stage 2''': Sign </br>
'''Output''': Account holder produces <math>\chi</math>

* Account holder generates <math>r</math>.
* For <math>i = 1, 2, ...n</math>:
** Account holder prepares <math>|\psi^{(i)}\rangle = f(k \mid\mid id\mid\mid r\mid\mid M\mid\mid i)</math>.
** Account holder encodes <math>|\psi^{(i)}\rangle</math> with <math>|\phi^{(i)}\rangle_{GHZ}</math> by combining <math>|\psi^{(i)}\rangle</math> with <math>|\phi^{(i)}\rangle_{A_1}</math> and performing a bell measurement on the two.
<div style="text-align: center;"><math>
|\phi^{(i)}\rangle = |\psi^{(i)}\rangle \otimes |\phi^{(i)}\rangle_{GHZ}
</math></div>
** Based on the measurement, account holder performs the suitable error correction, by applying the corresponding Pauli matrix, on <math>|\phi^{(i)}\rangle_{A_1}</math>:
<div style="text-align: center;"><math>
|\Phi^{(+)}\rangle \xrightarrow{} I,
|\Phi^{(-)}\rangle \xrightarrow{} \sigma_z,
|\Psi^{(+)}\rangle \xrightarrow{} \sigma_x,
|\Phi^{(-)}\rangle \xrightarrow{} \sigma_y,
</math></div>
* Account holder signs the serial number <math>s</math> using <math>\Pi</math>, as <math>\sigma \xleftarrow{} Sign_{sk}(s)</math>.
* Account holder produces <math>\chi</math>.

'''Stage 3''': Verify </br>
'''Output''': Cheque gets accepted or the process is aborted and the cheque is destroyed.

* Third party produces the <math>\chi</math> at a valid branch of the bank.
* Branch communicates with main branch of the bank and checks validity of <math>(id, s)</math> and runs <math>{Vrfy}_{pk}(\sigma, s)</math>.
** If invalid:
*** Bank aborts the process and destroys the cheque.
** else:
*** Bank continues the verification process.
* Main branch of the bank performs the measurement in Hadamard basis on its copy of <math>|\phi^{(i)}\rangle_{B}</math> and obtains outcome <math>|+\rangle</math> or <math>|-\rangle</math>.
* Main branch communicates this result with the local branch.
* For <math>i = 1, 2, ...n</math>:
** Based on outcome, branch performs the corresponding Pauli matrix operation on <math>|\phi^{(i)}\rangle_{A_2}</math> to recover <math>|\Psi^{(i)}\rangle</math> : <math>|+\rangle \xrightarrow{} I, |-\rangle \xrightarrow{} \sigma_z</math>
* For <math>i = 1, 2, ...n</math>:
** Bank computes <math>|\Psi^{,(i)}\rangle </math>, where,
<div style="text-align: center;"><math>
|\Psi^{,(i)}\rangle = f(k \mid\mid id\mid\mid r\mid\mid M\mid\mid i)
</math></div>
* For <math>i = 1, 2, ...n</math>:
** Bank performs swap test on <math>|\Psi^{(i)}\rangle</math> and <math>|\Psi^{,(i)}\rangle</math>.
*** If <math>\langle\Psi^{(i)}|\Psi^{,(i)}\rangle \geq \kappa</math>:
**** Bank aborts the process and destroys the cheque.
*** else:
**** Bank continues the verification process.
* Bank accepts the cheque.

==Further information==

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Quantum Cheque

2020-01-28T13:37:17Z

Rhea: /* Notation */

[https://link.springer.com/article/10.1007/s11128-016-1273-4 Quantum Cheque] is a private-key quantum money scheme that allows trusted banks to issue perfectly secure and verifiable quantum cheque books to account holders and also enables them to undertake monetary transactions with other parties. The quantum cheque is secure against [[non-signalling]] adversaries and cannot be counterfeited.

'''Tags:''' Private-Key Quantum Money, [[Quantum Digital Signature]], [[Quantum Key Distribution]], Quantum One-Way Function.

== Assumptions ==
* The protocol assumes the account holder and the bank, to be honest. The bank is a trusted party, however, the branches may or may not be trusted.
* This protocol assumes perfect state preparations, transmissions, and measurements.
* The protocol takes the assumption that the Quantum Digital Signature and the Quantum key distribution schemes are unconditionally secure.
* The digital signature scheme must satisfy the following security conditions of unforgeability and non-repudiation

==Outline==

This protocol allows a quantum cheque to be issued using quantum cheque books to the bank customers. The customers can then carry forward transactions in a perfectly secure manner and these cheques can be en-cashed after being verified by the trusted bank or its branches, that communicate with the main branch securely. The quantum cheque follows all the properties of a classical cheque - verifiable by a trusted bank, cannot be disavowed by the issuer and cannot be counterfeited by an adversary. </br>

The entire bank transaction process can be divided into three stages, Gen, where the cheque book is generated for the account holder, Sign, where the account holder prepares a cheque and issues it to the third party and Verify, where the third party en-cashes the cheque depending upon its validity.

* '''Gen'''
** In this stage, the bank first creates a cheque book and a key for the account holder.
** Then, the bank and the account holder create a shared key using [[Quantum Key Distribution]]. Both parties agree upon the [[Quantum Digital Signature]]. The account holder stores his private key safely with himself and shares the public key with the bank.
** The bank then prepares <math>n</math> GHZ triplet states (link to page) and stores only the third entangled qubit of every GHZ in the database, while handing over the first two qubits of every GHZ state to the account holder. Along with this, the bank also creates and shares a corresponding unique serial number for this cheque.
** Finally, the account holder has stored his identity, shared key, private key, public key, serial number and first two entangled qubits of every GHZ triplet state whereas the bank has stored each account holder's identity, shared key, public key, serial number and third entangled qubit of every GHZ triplet states in its database.<br></br>
* '''Sign'''
** In this stage, the issuer's key and amount to be signed is taken as input to produce a quantum state which acts like a cheque.
** Each account holder prepares <math>n</math> quantum states to issue a check worth some amount <math>M</math>. This quantum state is prepared by the quantum one-way function (Link to the page) which takes the shared key, identity of the account holder, a random number and the amount of money <math>M</math> to be signed as the input. Every state from these <math>n</math> quantum states is individually encoded with one of the account holder's entangled qubit of the GHZ states using Quantum Secret Sharing: Splitting of quantum information (Link to the page). The account holder then signs the serial number with their digital signature.
** The quantum cheque thus produces by the account holder contains the information - the identity of the account holder, serial number, generated a random number, digital signature, amount of money signed and the other entangled qubit of the GHZ state.<br></br>
* '''Verify'''
** In this method, the quantum cheque is verified by the trusted bank or its branches and the validity is checked.
** When the quantum cheque is produced at any valid branches of the bank by a third party, the information is securely communicated to the main branch. At the branch, an initial verification is carried by considering the identity of the issuer of the cheque, the serial number of the cheque and the digital signature of the issuer. If the cheque is valid, the verification process is continued, otherwise, the cheque is destroyed and the process is discontinued.
** The main branch performs a measurement on its copy of third entangled qubit of every GHZ triplet state which was stored in the database and securely communicates this result to the branch.
** The branch recovers the quantum state that was prepared by the account holder to issue the cheque, using the information received from the main branch. The branch also computes this same quantum states using the stored account holder's information like the shared key, the identity of the account holder, a random number and the amount of money <math>M</math> as input for verification. A swap test(Link to the page) is performed on both these states and if the cheque is only accepted if this test passes, else it is destroyed and aborted.

This scheme is proven to be impossible to counterfeit and impossible to repudiate. The quantum cheque is a quantum state and thus it cannot be copied or stolen by any eavesdropper, ensuring that only one copy of the quantum cheque exists.

==Requirements==
* Secure quantum channel to connect the branches of the bank to the main branch.
* Secure quantum channel to connect the bank to the account holder and to connect any other third party to the bank.
* This protocol required quantum memory to store issuer's quantum state of the cheque if the protocol is not running in real time.
* Private database for both account holder and bank.
* Measurement devices for the account holder and the main branch of the bank.

==Knowledge Graph==

{{graph}}
==Properties==

* In the signing process, the quantum one-way function used to create the cheque for the account holder is assumed to take polynomial time to compute and is hard to invert.
* In the verification process, the bank sets a thresholding security parameter <math>\kappa</math>. The swap test is passed if <math>\{ \langle\psi^{(i)}|\psi^{,(i)}\rangle \geq \kappa\}_{i=1:l}</math>
* No quantum memory would be required for the account holder to store the quantum check if the transaction is occurring in real time.
* This protocol can be realized, efficiently, with few qubit systems, without compromising on the security
* Security: This protocol is impossible to counterfeit and non-repudiation by signatory is impossible here.

==Notation==
* <math>k</math>: Shared key between account holder and bank where <math>k \in \{0,1\}^L</math>.
* <math>\Pi</math>: [[Quantum Digital Signature]] scheme, where <math>\Pi=(Gen, Sign, Vrfy)</math>.
* <math>id</math>: Account holder's identity.
* <math>pk</math>: Account holder's public key.
* <math>sk</math>: Account holder's private key.
* <math>s</math>: Unique serial number where <math>s \in \{0,1\}^{l(n)}</math>.
* <math>{|\phi^{(i)}\rangle_{GHZ}}</math>: <math>n</math> GHZ triplet states where,
<div style="text-align: center;"><math>|\phi^{(i)}\rangle_{GHZ} = \frac{1}{\sqrt{2}}(|0^{(i)}\rangle_{A_1}|0^{(i)}\rangle_{A_2}|0^{(i)}\rangle_{B} + |1^{(i)}\rangle_{A_1}|1^{(i)}\rangle_{A_2}|1^{(i)}\rangle_{B})</math></div>
* <math>{|\phi^{(i)}\rangle_{A_1}}</math>: First entangled qubit from every GHZ triplet state which is given to account holder.
* <math>{|\phi^{(i)}\rangle_{A_2}}</math>: Second entangled qubit from every GHZ triplet state which is given to account holder.
* <math>{|\phi^{(i)}\rangle_{B}}</math>: First entangled qubit from every GHZ triplet state which stays with bank.
* <math>r</math>: Generated random number where <math>r \in \{0,1\}^L</math>.
* <math>M</math>: Amount of money account holder signs on the cheque.
* <math>f</math>: Quantum one way function.
* <math>x\mid\mid</math>y: concatenation of two bit strings <math>x</math> and <math>y</math>.
* <math>{|\psi^{(i)}\rangle}</math>: quantum state prepared by the account where,
<div style="text-align: center;"><math>|\psi^{(i)}\rangle = f(k \mid\mid id\mid\mid r\mid\mid M\mid\mid i)</math></div>. This quantum state is also denoted as <math>\alpha_i|0\rangle + \beta_i|1\rangle</math>.
* <math>|\Phi^\pm\rangle</math>: Bell state <math>\frac{|00\rangle \pm |11\rangle}{\sqrt{2}}</math>
* <math>|\Psi^\pm\rangle</math>: Bell state <math>\frac{|10\rangle \pm |01\rangle}{\sqrt{2}}</math>
* <math>\sigma</math>: In <math>\chi</math>, output of signing with <math>s</math>.
* <math>\chi</math>: Quantum cheque where,
<div style="text-align: center;"><math>
\chi = (id, s, r, \sigma, M, \{{|\phi^{(i)}\rangle_{A_1}}\}_{i=1:n})
</math></div>
* <math>\kappa</math>: Threshold constant set as a security parameter by the bank in the swap test.
* <math>|\psi^{,(i)}\rangle</math>: Quantum state prepared by the bank for verification.

==Protocol Description==

'''Stage 1''': Gen</br>
'''Output''': Account holder now holds <math>(id, pk, sk, k, s, \{|\phi^{(i)}\rangle_{A_1}|\phi^{(i)}\rangle_{A_2}\}_{i=1:n})</math> and Bank now holds <math>(id, pk, k, s, {\|\phi^{(i)}\rangle_{B}\}_{i=1:n})}</math> in their private databases.

* Account holder and Bank create <math>k</math>.
* Account holder and Bank agree on <math>\Pi</math>.
* Account holder submits <math>pk</math> to Bank.
* Account holder secretly stores <math>sk</math>.
* For <math>i = 1, 2, ...n</math>:
** Bank generates GHZ triple state <math>|\phi^{(i)}\rangle_{GHZ}</math>
** Bank stores <math>|\phi^{(i)}\rangle_{B}</math> in their private database.
** Bank gives <math>|\phi^{(i)}\rangle_{A_1}|\phi^{(i)}\rangle_{A_2}</math> to account holder.
* Bank prepares <math>s</math> and shares it with the account holder.
* For <math>i = 1, 2, ...n</math>:
** Account holder stores <math>|\phi^{(i)}\rangle_{A_1}|\phi^{(i)}\rangle_{A_2}</math> privately.
* Account holder now holds <math>(id, pk, sk, k, s, \{|\phi^{(i)}\rangle_{A_1}|\phi^{(i)}\rangle_{A_2}\}_{i=1:n})</math>
* Bank now holds <math>(id, pk, k, s, {|\phi^{(i)}\rangle_{B}\}_{i=1:n})}</math>

'''Stage 2''': Sign </br>
'''Output''': Account holder produces <math>\chi</math>

* Account holder generates <math>r</math>.
* For <math>i = 1, 2, ...n</math>:
** Account holder prepares <math>|\psi^{(i)}\rangle = f(k \mid\mid id\mid\mid r\mid\mid M\mid\mid i)</math>.
** Account holder encodes <math>|\psi^{(i)}\rangle</math> with <math>|\phi^{(i)}\rangle_{GHZ}</math> by combining <math>|\psi^{(i)}\rangle</math> with <math>|\phi^{(i)}\rangle_{A_1}</math> and performing a bell measurement on the two.
<div style="text-align: center;"><math>
|\phi^{(i)}\rangle = |\psi^{(i)}\rangle \otimes |\phi^{(i)}\rangle_{GHZ}
</math></div>
** Based on the measurement, account holder performs the suitable error correction, by applying the corresponding Pauli matrix, on <math>|\phi^{(i)}\rangle_{A_1}</math>:
<div style="text-align: center;"><math>
|\Phi^{(+)}\rangle \xrightarrow{} I,
|\Phi^{(-)}\rangle \xrightarrow{} \sigma_z,
|\Psi^{(+)}\rangle \xrightarrow{} \sigma_x,
|\Phi^{(-)}\rangle \xrightarrow{} \sigma_y,
</math></div>
* Account holder signs the serial number <math>s</math> using <math>\Pi</math>, as <math>\sigma \xleftarrow{} Sign_{sk}(s)</math>.
* Account holder produces <math>\chi</math>.

'''Stage 3''': Verify </br>
'''Output''': Cheque gets accepted or the process is aborted and the cheque is destroyed.

* Third party produces the <math>\chi</math> at a valid branch of the bank.
* Branch communicates with main branch of the bank and checks validity of <math>(id, s)</math> and runs <math>{Vrfy}_{pk}(\sigma, s)</math>.
** If invalid:
*** Bank aborts the process and destroys the cheque.
** else:
*** Bank continues the verification process.
* Main branch of the bank performs the measurement in Hadamard basis on its copy of <math>|\phi^{(i)}\rangle_{B}</math> and obtains outcome <math>|+\rangle</math> or <math>|-\rangle</math>.
* Main branch communicates this result with the local branch.
* For <math>i = 1, 2, ...n</math>:
** Based on outcome, branch performs the corresponding Pauli matrix operation on <math>|\phi^{(i)}\rangle_{A_2}</math> to recover <math>|\Psi^{(i)}\rangle</math> : <math>|+\rangle \xrightarrow{} I, |-\rangle \xrightarrow{} \sigma_z</math>
* For <math>i = 1, 2, ...n</math>:
** Bank computes <math>|\Psi^{,(i)}\rangle </math>, where,
<div style="text-align: center;"><math>
|\Psi^{,(i)}\rangle = f(k \mid\mid id\mid\mid r\mid\mid M\mid\mid i)
</math></div>
* For <math>i = 1, 2, ...n</math>:
** Bank performs swap test on <math>|\Psi^{(i)}\rangle</math> and <math>|\Psi^{,(i)}\rangle</math>.
*** If <math>\langle\Psi^{(i)}|\Psi^{,(i)}\rangle \geq \kappa</math>:
**** Bank aborts the process and destroys the cheque.
*** else:
**** Bank continues the verification process.
* Bank accepts the cheque.

==Further information==

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>

Certification Library

2019-11-20T14:16:43Z

Rhea:

{| class="wikitable"
!width="40%"|Technique
!width="100%"|Protocols
|-
|rowspan="1"|[[Cross Entropy Benchmarking]]||[[Cross Entropy Benchmarking]]
|-
|rowspan="1"|[[Cycle Benchmarking]]||[[Cycle Benchmarking]]
|-
|rowspan="1"|[[Hamiltonian and Phase Estimation]]||[[Hamiltonian and Phase Estimation]]
|-
|rowspan="1"|[[Fidelity Estimation]]||[[Direct Fidelity Estimation]]
|-
|rowspan="1"|[[Fidelity Witnessing]]||[[Fidelity witnesses for fermionic quantum simulations]]
|-
|rowspan="2"|[[Process Tomography]]||[[Full Quantum Process Tomography with Linear inversion]]
|-
|[[Quantum Gate Set Tomography]]
|-
|rowspan="3"|[[Randomised Benchmarking]]||[[Interleaved Randomised Benchmarking]]
|-
|[[Purity Benchmarking]]
|-
|[[Standard Randomised Benchmarking]]
|-
|rowspan="7"|[[State Tomography]]||[[Compressed Sensing Tomography]]
|-
|[[Full Quantum state tomography with Linear Inversion]]
|-
|[[Full Quantum state tomography with Maximum Likelihood Estimation]]
|-
|[[Full Quantum state tomography with Bayesian mean estimation (BME)]]
|-
|[[Full Quantum state tomography using confidence regions]]
|-
|[[Matrix Product State tomography]]
|-
|[[Tensor Network Tomography]]
|-
|rowspan="1"|[[Quantum Volume Estimation]]||[[Quantum Volume Estimation]]
|-