Interleaved Randomised Benchmarking

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: Certification Protocol, 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 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} 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

• 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} : Depolarizing parameter
• 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 d} : Dimension of Hilbert space
• $F_{avg}$: Average fidelity, 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_{avg} = p + \frac{1-p}{d}}
• 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 r} : Average error rate, 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 r = 1- F_{avg}, r = \frac{(d-1)(1-p)}{d}}
• $m$: Selected sequence length
• 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 K_m} : Total randomly selected sequence 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 m} sequence length
• ClifFailed 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} : Clifford group
• CFailed 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 _i} : Random element of Clifford group
• 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 S_{(i_1, ...,i_m)}} = 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 S_{\mathbf{i_m}}} : Random sequence of operations of length $m$
• $M$: Number of different data points to get the error model
• $\Lambda _{i,j}$: Implementation of C$_{i}$ at time j (1 $\leq$ j $\leq$ M) results in this error map. $\Lambda _{i,1},...,\Lambda _{i,M}$ are the different time-dependent noise operators affecting CFailed 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 _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 |\psi\rangle} : initial 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 E_{\psi}} : POVM element which takes into account the measurement error.
• $F_{seq}(m,\psi )=Tr[E_{\psi }S_{\mathbf {i_{m}} }(\rho _{\psi })]$: Survival probability of a sequence. 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_\psi} is a quantum state that takes into account errors in preparing 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 \langle \psi |\psi \rangle}
• 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_g^{(0)}(m, |\psi\rangle)} : Averaged sequence fidelity for gate and time independent error model
• 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_g^{(1)}(m, |\psi\rangle)} : Averaged sequence fidelity for gate and time dependent error model. In this model, the parameter 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 (q-p^2)} is a measure of the degree of gate-dependence in the error.
• 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_0, B_0} : 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
• $A_{1},B_{1},C_{1}$: 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.
• 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 R_{m+1}} : 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 \frac{1}{|Clif_n|}\sum_i\Lambda_{i, m+1} \otimes (C_i \otimes \Lambda \otimes C_i^{\dagger})}

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 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} 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: 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 r}

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

Further Information

• Fitting models are described and derived as seen in E. Mageson et al. The coefficients derived are:
  • 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_0} = TrFailed 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 [E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]}
  • 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 B_0} = TrFailed 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 [E_\psi \Lambda(\frac{\mathbb{1}}{d})]}
  • 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_1} = TrFailed 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 [E_\psi \Lambda(\frac{Q_1\rho_\psi}{p} - \rho_\psi + \frac{(p-1)\mathbb{1}}{pd})]} + TrFailed 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 [E_\psi R_{m+1}(\frac{\rho_\psi}{p} - \frac{\mathbb{1}}{pd})]}
  • 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 B_1} = TrFailed 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 [E_\psi R_{m+1}(\frac{\mathbb{1}}{d})]}
  • 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 C_1} = TrFailed 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 [E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]}
• The case where Randomized benchmarking fails: Suppose the noise is time dependent and for each 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 i, \Lambda_i = C_i^{\dagger}} . 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 F_g(m, \psi) = 1} for every 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} even though there is a substantial error on each 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 C_i} and so benchmarking fails.
• 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 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 K} sequences of Clifford elements where the first Clifford 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 C_{i_1}} in each sequence is chosen uniformly at random from ClifFailed 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} , the second is always chosen to be 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 C} (gate of interest), and alternate between uniformly random Clifford elements and deterministic 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 C} up to 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 m^{th}} random gate.
  • 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 (m+1)^{th}} gate is chosen to be the inverse of the composition of the first 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} random gates and interlaced 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 C} 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 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_c} , which replaces 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} .
  • The new gate error is calculated 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 r_c = \frac{(d-1)(1-p_c/p)}{d}}
• 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

*contributed by Rhea Parekh