Editing Interleaved Randomised Benchmarking (section)

==Procedure Description==

'''Step 1''': Standard Randomised Benchmarking

'''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)} \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>
** 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>

'''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>