Editing Interleaved Randomised Benchmarking (section)

==Outline==
[[Standard Randomised Benchmarking]] method involves applying many random sequences of gates of varying lengths to a standard initial state. Each sequence ends with a randomized measurement that determines whether the correct final state was obtained. The average computationally relevant error per gate is obtained from the increase in error probability of the final measurements as a function of sequence length. The random gates are taken from the [[Clifford group]]. The restriction to the Clifford group ensures that the measurements can be of one-qubit Pauli operators that yield at least one deterministic one-bit answer in the absence of errors.

The multi-qubit RB protocol described in Standard Randomised Benchmarking is restricted to benchmark only the full [[Clifford group]] on <math>n</math> qubits.  While this provides a significant step towards scalable benchmarking of a quantum information processor, it is desirable in many cases to benchmark individual gates in Clifford group rather than the entire set. Interleaving randomised benchmarking is a protocol which consists of interleaving random gates between the gate of interest, which is used to estimate the average error of individual quantum computational gates.

To benchmark a specific Clifford element (an individual gate), the following steps are involved:

'''Step 1''': Implement [[Standard Randomised Benchmarking]] to get a model for the fidelity and to calculate the average error rate

* 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 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 then used to calculate the average survival probability for the sequence length.
* 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 depolarizing parameter and sequence fidelity and also to calculate the average error rate which is the metric for randomized benchmarking.

'''Step 2''': Procedure to estimate the new sequence fidelity by including the Clifford element to be benchmarked in the sequence

* Now, for a random fixed sequence length, choose a sequence where the first Clifford element is selected uniformly at random from the Clifford group and the second element is always chosen to be the specific Clifford element we want to benchmark.
* Final gate is chosen to be the inverse of the composition mentioned in the step above. 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 then used to calculate the average survival probability for the sequence length.
* The same procedure is repeated for multiple different randomly selected sequence lengths.
* 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.

'''Step 3''': Estimate the gate error of the selected Clifford element to be benchmarked

* From the values obtained for the depolarizing parameter and the new depolarizing parameter, using the average gate fidelity, a point estimate is obtained for the gate error. The gate error would lie in a certain range of this estimate. One interpretation of this error is that it arises from imperfect random gates.