# Standard Randomised Benchmarking

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: 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

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

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

## 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 ${\displaystyle p}$ 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: ${\displaystyle r}$

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

## Further Information

• Fitting models are described and derived as seen in E. Mageson et al. The coefficients derived are:
• ${\displaystyle A_{0}}$ = Tr${\displaystyle [E_{\psi }\Lambda (\rho _{\psi }-{\frac {\mathbb {1} }{d}})]}$
• ${\displaystyle B_{0}}$ = Tr${\displaystyle [E_{\psi }\Lambda ({\frac {\mathbb {1} }{d}})]}$
• ${\displaystyle A_{1}}$ = Tr${\displaystyle [E_{\psi }\Lambda ({\frac {Q_{1}\rho _{\psi }}{p}}-\rho _{\psi }+{\frac {(p-1)\mathbb {1} }{pd}})]}$ + Tr${\displaystyle [E_{\psi }R_{m+1}({\frac {\rho _{\psi }}{p}}-{\frac {\mathbb {1} }{pd}})]}$
• ${\displaystyle B_{1}}$ = Tr${\displaystyle [E_{\psi }R_{m+1}({\frac {\mathbb {1} }{d}})]}$
• ${\displaystyle C_{1}}$ = Tr${\displaystyle [E_{\psi }\Lambda (\rho _{\psi }-{\frac {\mathbb {1} }{d}})]}$
• The case where Randomized benchmarking fails: Suppose the noise is time dependent and for each ${\displaystyle i,\Lambda _{i}=C_{i}^{\dagger }}$. Then ${\displaystyle F_{g}(m,\psi )=1}$ for every ${\displaystyle m}$ even though there is a substantial error on each ${\displaystyle C_{i}}$ 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 ${\displaystyle K}$ sequences of Clifford elements where the first Clifford ${\displaystyle C_{i_{1}}}$ in each sequence is chosen uniformly at random from Clif${\displaystyle _{n}}$, the second is always chosen to be ${\displaystyle C}$(gate of interest), and alternate between uniformly random Clifford elements and deterministic ${\displaystyle C}$ up to the ${\displaystyle m^{th}}$ random gate.
• The ${\displaystyle (m+1)^{th}}$ gate is chosen to be the inverse of the composition of the first ${\displaystyle m}$ random gates and interlaced ${\displaystyle C}$ 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 ${\displaystyle p_{c}}$, which replaces ${\displaystyle p}$.
• The new gate error is calculated as ${\displaystyle r_{c}={\frac {(d-1)(1-p_{c}/p)}{d}}}$
• 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