Editing Standard Randomised Benchmarking (section)

==Further Information==
* Fitting models are described and derived as seen in [https://arxiv.org/abs/1109.6887v2 E. Mageson et al]. The coefficients derived are:
** <math>A_0</math> = Tr<math>[E_\psi \Lambda(\rho_\psi - \frac{\mathbb{1}}{d})]</math>
** <math>B_0</math> = Tr<math>[E_\psi \Lambda(\frac{\mathbb{1}}{d})]</math>
** <math>A_1</math> = Tr<math>[E_\psi \Lambda(\frac{Q_1\rho_\psi}{p} - \rho_\psi + \frac{(p-1)\mathbb{1}}{pd})]</math> + Tr<math>[E_\psi R_{m+1}(\frac{\rho_\psi}{p} - \frac{\mathbb{1}}{pd})]</math>
** <math>B_1</math> = Tr<math>[E_\psi R_{m+1}(\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.
* [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.