Editing Standard Randomised Benchmarking

[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>