Standard Randomize Benchmarking

An approach to obtain the average gate fidelity is to perform randomized benchmarking [6]. The key idea is that if we average the error process over the uniform space of unitaries the result is a depolarizing channel that maps any pure state to the maximally mixed state. Such uniform space of unitaries is known as Haar measure. It can be shown [6] that the average induced error is proportional to the depolarization probability. However, this approach is inefficient because we sample randomly from the Haar measure. A simplification was proposed in [12] by restricting the unitaries to the Clifford group, which consists of unitary rotations mapping the group of Pauli operators in itself. Among the advantages of such group are the fact of the number of Clifford gates is finite given the Hilbert space and being a group we can easily found the inverse within the group. The generic procedure to perform a randomized benchmarking is the following:

1. initialize the system in ground state

2. for each sequence length $m$ draw sequence of Clifford group elements

3. calculate inverse gate

4. measure sequence and inverse gate

5. repeat the process for multiple sequence of same length and varying the length

The previous approach works because it has been shown [16] that randomization with Clifford gates provides again a depolarized noise channel

(1)$$\rho \to \frac{d}{2}I+(1-d)\rho$$

with depolarization probability $d$. If we follow the previous procedure and we measure the survival probability, i.e. the probability of measuring the qubit in $|0⟩$, for different sequence length $m$ we expect the following behavior

(2)$$F(m)=A{p}^m+B$$

where $1-p$ is the rate of depolarization while $A$ and $B$ capture state preparation and measurement errors. Finally, we can extract the average error per Clifford as

(3)$${ϵ}_{\text{Clifford}}=\frac{1-p}{1-2^{-n}}$$

where $n$ is the number of qubits. The error per gate can be derived by dividing the Clifford error by the physical gates per Clifford which usually is 1.875. One of the main feature of RB is the possibility to estimate the gate fidelity alone without taking into account both state preparation and measurement errors which can be computed using the $A$ and $B$ terms in Eq. 2.

Parameters

class qibocal.protocols.randomized_benchmarking.standard_rb.StandardRBParameters(depths: Union[list, Depthsdict], niter: int, uncertainties: Optional[float] = None, unrolling: bool = False, seed: Optional[int] = None, nshots: int = 10)

Standard Randomized Benchmarking runcard inputs.

depths: Union[list, Depthsdict]

A list of depths/sequence lengths.

If a dictionary is given the list will be build.

niter: int

Sets how many iterations over the same depth value.

uncertainties: Optional[float] = None

Method of computing the error bars of the signal and uncertainties of the fit.

If None, it computes the standard deviation. Otherwise it computes the corresponding confidence interval. Defaults None.

unrolling: bool = False

If True it uses sequence unrolling to deploy multiple circuits in a single instrument call.

Defaults to False.

seed: Optional[int] = None

A fixed seed to initialize np.random.Generator.

If None, uses a random seed. Defaults is None.

nshots: int = 10

Just to add the default value.

hardware_average: bool = False

By default hardware average will be performed.

relaxation_time: float

Wait time for the qubit to decohere back to the gnd state.

Example

It follows a runcard where we execute a standard RB.

- id: standard rb
  operation: standard_rb
  parameters:
    depths: [1,5,10,20,50,100]
    niter: 20
    nshots: 100

The expected output is the following: