Editing Standard Randomised Benchmarking
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[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. | [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 | '''Tags:''' [[:Category: Certification protocol|Certification Protocol]], [[:Category: Average gate fidelity|Average gate fidelity]], Clifford group | ||
==Assumptions== | ==Assumptions== | ||
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This method consists of the following steps: | This method consists of the following steps: | ||
* A fixed sequence length is selected | * 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 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). | * 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 | * 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 | * The same procedure is repeated for multiple different 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. | * 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. | ||
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* C<math>_i</math>: Random element of 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>S_{(i_1, ...,i_m)}</math> = <math>S_{\mathbf{i_m}}</math>: Random sequence of operations of length <math>m</math> | ||
* <math>M</math>: | * <math>M</math>: Maximum sequence length of applying Clifford group Clif<math>_n</math> | ||
* <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>\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>|\psi\rangle</math>: initial state | ||
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* '''Figure of merit''': average error rate, average gate fidelity | * '''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 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. | * The random gates are picked from the Clifford group. However in the case of [[interleaved randomized benchmarking]] | ||
* For noise estimation, the uniform probability distribution over Clifford group comprises a [[unitary 2-design]]. | * 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. | * This protocol provides a scalable method for benchmarking the set of Clifford gates. | ||
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'''Output''': Figure of merit: <math>r</math> | '''Output''': Figure of merit: <math>r</math> | ||
* For <math>1, 2, ..., M</math>: | * For <math>m = 1, 2, ..., M-1</math>: | ||
** For <math>k = 1, 2, ..., K_m</math> sequences: | ** For <math>k = 1, 2, ..., K_m</math> sequences: | ||
*** For <math>j = 1, 2 ..., m+1</math>: | *** For <math>j = 1, 2 ..., m+1</math>: | ||
**** If <math>j == m+1</math>, apply inverse operator of previous operations | **** If <math>j == m+1</math>, apply inverse operator of previous operations | ||
**** else, apply random operation C<math>_i</math> | **** else, apply random operation C<math>_i</math> | ||
*** Thus, <math>S_{\mathbf{i_m}} = \bigotimes^{m+1}_{j | *** Thus, <math>S_{\mathbf{i_m}} = \bigotimes^{m+1}_{j+1} (\Lambda_{(i_j, j)} C_i)</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> | *** 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> | ** 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> | ||
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* 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. | * 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: | * [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 | ** 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$_n$, 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 <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 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>. |