Editing Interleaved Randomised Benchmarking
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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]], [[Randomised Benchmarking]], Clifford group | '''Tags:''' [[:Category: Certification protocol|Certification Protocol]], [[:Category: Average gate fidelity|Average gate fidelity]], [[Randomised Benchmarking]], Clifford group | ||
==Assumptions== | ==Assumptions== | ||
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* 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. | * 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. | ||
==Hardware Requirements== | ==Hardware Requirements== | ||
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==Notation== | ==Notation== | ||
* <math>p</math>: Depolarizing parameter | * <math>p</math>: Depolarizing parameter | ||
* <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<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>: 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>|\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>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''': | * '''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 | * The random gates are picked from the Clifford group. | ||
* 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. | |||
* 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== | ==Procedure Description== | ||
'''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)} | *** 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> | *** 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> | ||
==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. | ||
* [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. | * 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 | ||
* 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> | <div style='text-align: right;'>''*contributed by Rhea Parekh''</div> |