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, 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 $n$ 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.

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

• $p$: Depolarizing parameter
• $p_{\bar {C}}$: New depolarizing parameter for the specific Clifford element to be benchmarked
• $d$: Dimension of Hilbert space
• $F_{avg}$: Average fidelity, $F_{avg}=p+{\frac {1-p}{d}}$
• $r$: Average error rate, $r=1-F_{avg},r={\frac {(d-1)(1-p)}{d}}$
• $m$: Selected sequence length
• $K_{m}$: Total randomly selected sequence of $m$ sequence length
• Clif$_{n}$: Clifford group
• C: Selected Clifford element to be benchmarked
• C$_{i}$: Random element of Clifford group
• $S_{(i_{1},...,i_{m})}$ = $S_{\mathbf {i_{m}} }$: Random sequence of operations of length $m$
• $\gamma$: Superoperator representing the sequence with alternating $C$
• $M$: 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 C$_{i}$.
• 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 \Lambda_{C}} : Associated noise operator of the Clifford element $C$
• $r_{C}^{est}$: The gate error of $\Lambda _{C}$
• $E$: Error range of $r_{C}^{est}$
• 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
• $E_{\psi }$: 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. $\rho _{\psi }$ is a quantum state that takes into account errors in preparing $\langle \psi |\psi \rangle$
• $F_{g}^{(0)}(m,|\psi \rangle )$: Averaged sequence fidelity for gate and time independent error model
• $F_{g}^{(1)}(m,|\psi \rangle )$: Averaged sequence fidelity for gate and time dependent error model. In this model, the parameter $(q-p^{2})$ is a measure of the degree of gate-dependence in the error.
• $F_{\bar {g}}^{(0)}(m,|\psi \rangle )$: New zeroth order averaged sequence fidelity for $C$
• $F_{\bar {g}}^{(1)}(m,|\psi \rangle )$: New first order Averaged sequence fidelity for $C$
• $A_{0},B_{0}$: 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.
• $R_{m+1}$: ${\frac {1}{|Clif_{n}|}}\sum _{i}\Lambda _{i,m+1}\otimes (C_{i}\otimes \Lambda \otimes C_{i}^{\dagger })$

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 $p$ one should always use the first order fitting model unless prior knowledge of the noise indicates that it is effectively gate-independent.

Procedure Description

Step 1: Standard Randomised Benchmarking

Output: Figure of merit: $r$

• For $1,2,...,M$:
  • Pick random sequence length $m$
  • For $k=1,2,...,K_{m}$ sequences:
    • For $j=1,2...,m+1$:
      • If $j==m+1$, apply inverse operator of previous operations
      • else, apply random operation C$_{i}$
    • Thus, $S_{\mathbf {i_{m}} }=\bigotimes _{j=1}^{m+1}(\Lambda _{(i_{j},j)}\circ C_{i_{j}})$ and $i_{m+1}$ is uniquely determined by $(i_{1},...,i_{m})$
    • Measure survival probability $Tr[E_{\psi }S_{\mathbf {i_{m}} }(\rho _{\psi })]$
  • Estimate average survival probability $Tr[E_{\psi }S_{\mathbf {K_{m}} }(\rho _{\psi })]$ over all $K_{m}$ sequences, where $S_{\mathbf {K_{m}} }={\frac {1}{K_{m}}}\sum _{i_{m}}S_{i_{m}}$
• Fit the results for the averaged sequence fidelity for all $m$ into the models:
  • For gate and time independent error model:
    • $F_{g}^{(0)}(m,|\psi \rangle )=A_{0}p^{m}+B_{0}$
  • For gate and time dependent error model:
    • $F_{g}^{(1)}(m,|\psi \rangle )=A_{1}p^{m}+B_{1}+C_{1}(m-1)(q-p^{2})p^{m-2}$
• $p$ is extracted from the model and $r$ is estimated, $r={\frac {(d-1)(1-p)}{d}}$

Step 2: Estimate gate error of selected Clifford element C

Input: C

Output: gate error of $\Lambda _{C}$: $r_{C}^{est}$

• For $1,2,...,M$:
  • Pick random sequence length $m$
  • For $k=1,2,...,K_{m}$ sequences:
    • For $j=1,2...,m+1$:
      • If $j==m+1$, apply inverse operator of previous operations
      • else If $j\%2==1$, apply random operation C$_{i}$
      • else, apply C
    • Thus $\gamma =\Lambda _{i_{m+1}}+$ C$_{i_{m+1}}(\bigotimes _{j=1}^{m+1}[C\circ \Lambda _{C}\circ \Lambda _{i_{j}}\circ C_{i_{j}}])$
    • Measure the survival probability $Tr[E_{\psi }\gamma _{\mathbf {i_{m}} }(\rho _{\psi })]$
  • Estimate average survival probability $Tr[E_{\psi }\gamma _{\mathbf {K_{m}} }(\rho _{\psi })]$ over all $K_{m}$ sequences, where $\gamma _{\mathbf {K_{m}} }={\frac {1}{K_{m}}}\sum _{i_{m}}\gamma _{i_{m}}$
• Fit the results for the averaged sequence fidelity for all $m$ into the models, to find $p_{\bar {C}}$:
  • For gate and time independent error model:
    • $F_{\bar {g}}^{(0)}(m,|\psi \rangle )=A_{0}p_{\bar {C}}^{m}+B_{0}$
  • For gate and time dependent error model:
    • $F_{\bar {g}}^{(1)}(m,|\psi \rangle )=A_{1}p_{\bar {C}}^{m}+B_{1}+C_{1}(m-1)(q-p_{\bar {C}}^{2})p_{\bar {C}}^{m-2}$
• Estimate $r_{C}^{est}={\frac {(d-1)(1-p_{\bar {C}}/p)}{d}}$
• $r_{C}^{est}$ lies in the range $[r_{C}^{est}-E,r_{C}^{est}+E]$, where $E=min({\frac {(d-1)[|p-p_{\bar {C}}/p|+(1-p)]}{d}},{\frac {2(d^{2}-1)(1-p)}{pd^{2}}}+{\frac {4{\sqrt {1-p}}{\sqrt {d^{2}-1}}}{p}})$

Further Information

• Fitting models are described and derived as seen in E. Mageson et al. The coefficients derived are:
  • $A_{0}$ = Tr$[E_{\psi }\Lambda (\rho _{\psi }-{\frac {\mathbb {1} }{d}})]$
  • $B_{0}$ = Tr$[E_{\psi }\Lambda ({\frac {\mathbb {1} }{d}})]$
  • $A_{1}$ = Tr$[E_{\psi }\Lambda ({\frac {Q_{1}\rho _{\psi }}{p}}-\rho _{\psi }+{\frac {(p-1)\mathbb {1} }{pd}})]$ + Tr$[E_{\psi }R_{m+1}({\frac {\rho _{\psi }}{p}}-{\frac {\mathbb {1} }{pd}})]$
  • $B_{1}$ = Tr$[E_{\psi }R_{m+1}({\frac {\mathbb {1} }{d}})]$
  • $C_{1}$ = Tr$[E_{\psi }\Lambda (\rho _{\psi }-{\frac {\mathbb {1} }{d}})]$
• The case where Randomized benchmarking fails: Suppose the noise is time dependent and for each $i,\Lambda _{i}=C_{i}^{\dagger }$. Then $F_{g}(m,\psi )=1$ for every $m$ even though there is a substantial error on each $C_{i}$ 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

*contributed by Rhea Parekh