# Randomized Benchmarking¶

## Introduction¶

Randomized benchmarking allows for extracting information about the fidelity of a quantum operation by exploiting twirling errors over an approximate implementation of the Clifford group [KL+08]. This provides the advantage that the fidelity can be learned without simulating the dynamics of a quantum system. Instead, benchmarking admits an analytic form for the survival probability for an arbitrary input state in terms of the strength $$p$$ of an equivalent depolarizing channel.

QInfer supports randomized benchmarking by implementing this survival probability as a likelihood function. This allows for randomized benchmarking to be used together with Sequential Monte Carlo, such that prior information can be incorporated and robustness to finite sampling can be obtained [GFC14].

Regardless of the order or interleaving mode, each model instance for randomized benchmarking yields 0/1 data, with 1 indicating a survival (measuring the same state after applying a gate sequence as was initially prepared). To use these models with data batched over many sequences, model instances can be augmented by BinomialModel.

## Zeroth-Order Model¶

The RandomizedBenchmarkingModel class implements randomized benchmarking as a QInfer model, both in interleaved and non-interleaved modes. For the non-interleaved mode, there are three model parameters, $$\vec{x} = (p, A_0, B_0)$$, given by [MGE12] as

$\begin{split}A_0 & := \Tr\left[E_\psi \Lambda\left(\rho_\psi - \frac{\ident}{d}\right)\right] \\ B_0 & := \Tr\left[E_\psi \Lambda\left(\frac{\ident}{d}\right)\right] \\ p & := (d F_\ave - 1) / (d - 1),\end{split}$

where $$E_\psi$$ is the measurement, $$\rho_\psi$$ is the state preparation, $$\Lambda$$ is the average map over timesteps and gateset elements, $$d$$ is the dimension of the system, and where $$F_\ave$$ is the average gate fidelity, taken over the gateset. The functions p and F convert back and forth between depolarizing parameters and fidelities.

An experiment parameter vector for this model is simply a specification of $$m$$, the length of the Clifford sequence used for that datum. Since RandomizedBenchmarkingModel represents 0/1 data, it is common to wrap this model in a BinomialModel:

>>> from qinfer import BinomialModel
>>> from qinfer import RandomizedBenchmarkingModel
>>> model = BinomialModel(RandomizedBenchmarkingModel(order=0, interleaved=False))
>>> expparams = np.array([
...    (100, 1000) # 1000 shots of sequences with length 100.
... ], dtype=model.expparams_dtype)


### Interleaved Mode¶

If one is interested in the fidelity of a single gate, rather than an entire gateset, then the gate of interest can be interleaved with other gates from the gateset to isolate its performance. In this mode, models admit an additional model and experiment parameter, $$\tilde{p}$$ and mode, respectively. The $$\tilde{p}$$ model parameter is the depolarizing strength of the twirl of the interleaved gate, such that the interleaved survival probability is given by

$\Pr(\text{survival} | \tilde{p}, p_{\text{ref}}, A_0, B_0; m, \text{interleaved}) = A_0 (\tilde{p} p_{\text{ref}})^m + B_0.$

Model instances for interleaved mode are constructed using the interleaved=True keyword argument:

>>> from qinfer.rb import RandomizedBenchmarkingModel
>>> model = RandomizedBenchmarkingModel(interleaved=True)