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Standard Randomised Benchmarking: Difference between revisions
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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 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 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 used to calculate the average survival probability for the sequence length. | * 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 sequence lengths. | * 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. | * 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>: 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 | ||
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'''Output''': Figure of merit: <math>r</math> | '''Output''': Figure of merit: <math>r</math> | ||
* For <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>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_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> | ||