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[https://arxiv.org/abs/1203.4550 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. | [https://arxiv.org/abs/1203.4550 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:''' [[:Category: Certification protocol|Certification Protocol | '''Tags:''' [[:Category: Certification protocol|Certification Protocol]], [[Randomised Benchmarking]], Clifford group | ||
==Assumptions== | ==Assumptions== | ||
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'''Step 3''': Estimate the gate error of the selected Clifford element to be benchmarked | '''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. | * 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== | ==Hardware Requirements== | ||
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==Notation== | ==Notation== | ||
* <math>p</math>: Depolarizing parameter | * <math>p</math>: Depolarizing parameter | ||
* <math>p_{\bar{C}}</math>: New depolarizing parameter for the specific Clifford element to be benchmarked | |||
* <math>d</math>: Dimension of Hilbert space | * <math>d</math>: Dimension of Hilbert space | ||
* <math>F_{avg}</math>: Average fidelity, <math>F_{avg} = p + \frac{1-p}{d}</math> | * <math>F_{avg}</math>: Average fidelity, <math>F_{avg} = p + \frac{1-p}{d}</math> | ||
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* <math>K_m</math>: Total randomly selected sequence of <math>m</math> sequence length | * <math>K_m</math>: Total randomly selected sequence of <math>m</math> sequence length | ||
* Clif<math>_n</math>: Clifford group | * Clif<math>_n</math>: Clifford group | ||
* C: Selected Clifford element to be benchmarked | |||
* 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>\gamma</math>: Superoperator representing the sequence with alternating <math>C</math> | |||
* <math>M</math>: Number of different data points to get the error model | * <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>\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_{C}</math>: Associated noise operator of the Clifford element <math>C</math> | |||
* <math>r^{est}_C</math>: The gate error of <math>\Lambda_{C}</math> | |||
* <math>E</math>: Error range of <math>r^{est}_C</math> | |||
* <math>|\psi\rangle</math>: initial state | * <math>|\psi\rangle</math>: initial state | ||
* <math>E_{\psi}</math>: POVM element which takes into account the measurement error. | * <math>E_{\psi}</math>: POVM element which takes into account the measurement error. | ||
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* <math>F_g^{(0)}(m, |\psi\rangle)</math>: Averaged sequence fidelity for gate and time independent error model | * <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>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>F_\bar{g}^{(0)}(m, |\psi\rangle)</math>: New zeroth order averaged sequence fidelity for <math>C</math> | |||
* <math>F_\bar{g}^{(1)}(m, |\psi\rangle)</math>: New first order Averaged sequence fidelity for <math>C</math> | |||
* <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_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>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. | ||
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==Properties== | ==Properties== | ||
* '''Figure of merit''': average error | * '''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 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. | ||
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==Procedure Description== | ==Procedure Description== | ||
'''Step 1''': Standard Randomised Benchmarking | |||
'''Output''': Figure of merit: <math>r</math> | '''Output''': Figure of merit: <math>r</math> | ||
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**** 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=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> | *** Thus, <math>S_{\mathbf{i_m}} = \bigotimes^{m+1}_{j=1} (\Lambda_{(i_j, j)} \circ 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> | *** 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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*** <math>F_g^{(1)}(m, |\psi\rangle) = A_1p^m + B_1 + C_1(m-1)(q-p^2)p^{m-2}</math> | *** <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> | * <math>p</math> is extracted from the model and <math>r</math> is estimated, <math>r = \frac{(d-1)(1-p)}{d}</math> | ||
'''Step 2''': Estimate gate error of selected Clifford element C | |||
'''Input''': C | |||
'''Output''': gate error of <math>\Lambda_{C}</math>: <math>r^{est}_C</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 If <math>j%2==1</math>, apply random operation C<math>_i</math> | |||
**** else, apply C | |||
*** Thus <math>\gamma = \Lambda_{i_{m+1}} +</math> C<math>_{i_{m+1}} (\bigotimes^{m+1}_{j=1}[C \circ \Lambda_C \circ \Lambda_{i_j} \circ C_{i_j}])</math> | |||
*** Measure the survival probability <math>Tr[E_{\psi}\gamma_{\mathbf{i_m}}(\rho_\psi)]</math> | |||
** Estimate average survival probability <math>Tr[E_{\psi}\gamma_{\mathbf{K_m}}(\rho_\psi)]</math> over all <math>K_m</math> sequences, where <math>\gamma_{\mathbf{K_m}} = \frac{1}{K_m}\sum_{i_m} \gamma_{i_m}</math> | |||
* Fit the results for the averaged sequence fidelity for all <math>m</math> into the models, to find <math>p_{\bar{C}}</math>: | |||
** For gate and time independent error model: | |||
*** <math>F_\bar{g}^{(0)}(m, |\psi\rangle) = A_0p_{\bar{C}}^m + B_0</math> | |||
** For gate and time dependent error model: | |||
*** <math>F_\bar{g}^{(1)}(m, |\psi\rangle) = A_1p_{\bar{C}}^m + B_1 + C_1(m-1)(q-p_{\bar{C}}^2)p_{\bar{C}}^{m-2}</math> | |||
* Estimate <math>r^{est}_C = \frac{(d-1)(1-p_{\bar{C}}/p)}{d}</math> | |||
* <math>r^{est}_C</math> lies in the range <math>[r^{est}_C-E, r^{est}_C+E]</math>, where <math>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})</math> | |||
==Further Information== | ==Further Information== | ||
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** <math>C_1</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \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. | * 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. | ||
* 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. | * 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. | ||
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* E. Mageson et al (2011) arXiv:1009.3639: multi-parameter model | * E. Mageson et al (2011) arXiv:1009.3639: multi-parameter model | ||
* Magesan et al. PRL (2012): Interleaved Randomized Benchmarking | * Magesan et al. PRL (2012): Interleaved Randomized Benchmarking | ||
<div style='text-align: right;'>''*contributed by Rhea Parekh''</div> | <div style='text-align: right;'>''*contributed by Rhea Parekh''</div> | ||