Full Quantum Process Tomography with Linear inversion

Quantum process tomography is a method used to characterize a physical process in an open quantum system. This method is similar to Quantum state tomography, but the goal here is to characterize a quantum gate instead of a state. The method described here performs quantum process tomography using linear inversion.

Tags: Certification protocol, Tomography, Quantum process density matrix reconstruction, Linear inversion

Assumptions

• Noise is assumed to be independent and identically distributed.
• The initial states and final measurements are known.
• Here the case of a “nonselective” quantum evolution, such as arises from uncontrolled interactions with an environment.

Outline

Quantum process tomography is used to characterise an unknown quantum dynamical process. The general way to describe state change or a process experienced by a quantum system is by using quantum operations. This quantum process operation is a linear map which completely describes the dynamics of the quantum system.

This linear map can be described by an operator sum representation, where the map is broken down into different operators acting on the system. These different operators completely describe the state changes of the system including any possible unitary operations. It is convenient to consider an equivalent description of the process using a fixed set of these operators, so as to relate them to measurable parameters.

This method consists of the following steps:

• For a system with the state space having Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} dimensions, A set of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N^2} linearly independent basis elements are selected and the corresponding quantum states are prepared.
• The quantum process is performed on each of these states and the output state is measured using quantum state tomography.
• These output states are expressed as a linear combination of the basis states. A mathematical relation is formed which is used to determine the different operators which describe the quantum process.

Hardware Requirements

• Measurement device.
• Quantum computational resources to perform the operation.

Notation

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} : Dimension of state space
• $\varepsilon$: Quantum process operator. This is the linear map which completely describes the dynamics of a quantum system, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho \xrightarrow[]{} \frac{\varepsilon(\rho)}{tr(\varepsilon(\rho))}} . The operator sum representation of $\varepsilon$ is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon(\rho) = \sum_{i}A_i\rho A^{\dagger}_i}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_i} : Operators acting on the system alone, yet they completely describe the state changes of the system, including any possible unitary operation.
• ${\tilde {A_{i}}}$: Fixed set of operators used to describe $\varepsilon$ which form a basis for the set of operators on the state space, so that $A_{i}=\sum _{m}a_{im}{\tilde {A_{m}}}$. This is done to related Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_i} to measurable operators.
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{im}} : Set of complex numbers
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_j} : A set of linearly independent basis elements for the space of N x N matrices. A convienent choice is the set of projectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |m\rangle\langle n|}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} : Classical error correlation matrix which is positive Hermitian by definition. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} is completely described by this. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi_{mn} = \sum_i a_{im}a_{in}^{*}}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_{jk}} : Parameter which can be determined from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon{(\rho_j)}}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta_{jk}^{mn}} : Complex numbers which can be determined by standard algorithms given the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{A_m}} operators and the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_j} operators. This is A Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N^4} x Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N^4} matrix with columns indexed $mn$ and rows indexed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ij} . Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{A_m}\rho_j\tilde{A_n^{\dagger}} = \sum_k \beta^{mn}_{jk} \rho_k} .
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa} : the generalized inverse for the matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta} , satisfying the relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta_{jk}^{mn} = \sum_{st, xy} \beta_{jk}^{st}\kappa^{xy}_{st}\beta^{mn}_{xy}}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U^{\dagger}} : Unitary matrix which diagonalizes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi}

Properties

• Figure of merit: Density Matrix of the quantum process
• Multiple copies of the quantum state are required in this method.
• The process Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} can be completely determined by $\chi$ once the set of operators Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{A_i}} has been fixed.
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} will contain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N^4 - N^2} parameters
• In the case of a “nonselective” quantum evolution, such as arises from uncontrolled interactions with an environment the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_i} operators satisfy an additional completeness relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_i A_i^{\dagger} A_i = I} . This relation ensures that the trace factor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle tr(\varepsilon(\rho))} is always equal to one, and thus the state change experienced by the system can be written Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho \xrightarrow{} \varepsilon(\rho)}
• Quantum state tomography has to be performed multiple times in this experiment.
• This method is extremely resource intensive.
• The main drawback of this technique is that sometimes the computed solution of the density matrix will not be a valid density matrix. This issue occurs when not enough measurements are made. Due to this drawback, the maximum likelihood estimation technique [link here] is preferred over this linear inversion technique.
• Another drawback of this method is that an infinite number of measurement outcomes would be required to give the exact solution.
• Entanglement fidelity can be evaluated with the knowledge of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_i} operators. This quantity can be used to measure how closely the dynamics of the quantum system under consideration approximates that of some ideal quantum system. This quantity can be determined robustly, because of its linear dependence on the experimental errors. Suppose the target quantum operation is a unitary quantum operation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\rho) = U\rho U^\dagger} and the actual quantum operation implemented experimentally is $\varepsilon$. Then the entanglement fidelity is $F_{e}(\rho ,U,\varepsilon )=\sum _{i}{|tr(U^{\dagger }A_{i}\rho )|}^{2}=\sum _{mn}\chi _{mn}tr(U^{\dagger }{\tilde {A_{m}}}\rho )tr(\rho {\tilde {A_{n}^{\dagger }}}U)$
• The minimum value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_e} over all possible states Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho} is a single parameter which describes how well the experimental system implements the desired quantum logic gate
• Minimum fidelity of the gate operation is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F = min_{|\psi \rangle} \langle\psi| U^{\dagger} \varepsilon (|\psi\rangle\langle\psi|) U|\psi\rangle}
• Quantum channel capacity which is a measure of the amount of quantum information that can be sent reliably using a quantum communications channel which is described by a quantum operation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} can also be defined.
• This procedure can also be used to determine the form of the Lindblad operator used in Markovian master equations of a certain form.
• Quantum operations can also be used to describe measurements. For each measurement outcome, $i$, there is a associated quantum operation $\varepsilon _{i}$. The corresponding state change is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho \xrightarrow{} \varepsilon_i(\rho)/tr(\varepsilon_i(\rho))} where the probability of the measurement outcome occurring is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_i = tr(\varepsilon_i(\rho))} . To determine the process the steps are the same, except the measurement has to be taken a large enough number of times that the probability pi can be reliably estimated. Next Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho^{'}_{j}} is determined using tomography, thus obtaining Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon_i(\rho_j) = tr(\varepsilon_i(\rho_j))\rho^{'}_{j}} for each input Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_j} . The other steps remain to same to estimate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon_i} .

Protocol Description

Input: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_j, j = 1, ..., N^2}

Output: Density matrix of the quantum process operator, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon}

• Select a fixed set of operators Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{A_i}} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_i = \sum_m a_{im} \tilde{A_m}}
• The operator sum representation is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon(\rho) = \sum_{i}A_i\rho A^{\dagger}_i}
• Hence, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon(\rho) = \sum_{mn}\tilde{A_m}\rho \tilde{A^{\dagger}_n}\chi_{mn}}
• For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_j = 1, 2, ..., N^2} :
  • The process Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} is performed on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_j}
  • The output state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon(\rho_j)} is measured using quantum state tomography
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon(\rho_j)} is expressed as a linear combination of basis states, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon(\rho_j) = \sum_k \lambda_{jk} \rho_k}
  • Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{A_m}\rho_j\tilde{A_n^{\dagger}} = \sum_k \beta^{mn}_{jk} \rho_k} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_k \sum_{mn} \chi_{mn} \beta^{mn}_{jk} \rho_k = \sum_k \lambda_{jk} \rho_k}
  • Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_k} is independent, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_{jk} = \sum_{mn} \beta^{mn}_{jk} \chi_{mn}}
• From there Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi_{mn}} is defined as, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi_{mn} = \sum_{jk} \kappa^{mn}_{jk}}
• Let the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U^{\dagger}} diagonalize Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi_{mn} =\sum_{xy} U_{mn}d_x\delta_{xy}U^{*}_{ny}}
• From there, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_i = \sqrt{d_i} \sum_j U_{ij} \tilde{A_j}}
• Thus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon(\rho)} can be determined using Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon(\rho) = \sum_{i}A_i\rho A^{\dagger}_i}

Further Information

• Maximum likelihood estimation (also known as MLE or MaxLik) is a popular technique that overcomes the problem that the naive matrix inversion procedure in QPT, when performed on real (i.e., inherently noisy) experimental data, typically leads to an unphysical process matrix.
• The standard approach of process tomography is grossly inaccurate in the case where the states and measurement operators used to interrogate the system are generated by gates that have some systematic error, a situation all but unavoidable in any practical setting. These errors in tomography cannot be fully corrected through oversampling or by performing a larger set of experiments. Hence gate set tomography was introduced.

Related Papers

• Chuang et al arXiv:quant-ph/9610001v1: Prescription for experimental determination of

the dynamics of a quantum black box

• J. F. Poyatos, J. I. Cirac, and P. Zoller, PhysRevLett.78.390: Complete Characterization of a Quantum Process: The Two-Bit Quantum Gate
• M.W. Mitchell, et al., Phys. Rev. Lett. 91, 120402 (2003)
• J. L. O’Brien et al arXiv:quant-ph/0402166v2: Quantum process tomography of a controlled-not gate
• S. T. Merkel, J. M. Gambetta, J. A. Smolin, S. Poletto, A. D. Corcoles, B. R.
• Johnson, C. A. Ryan, and M. Steffen, Phys. Rev. A 87, 062119 (2013).

*contributed by Rhea Parekh