Linear XEB fidelity estimator.

Estimates fidelity from ideal probabilities of observed bitstrings.

This estimator makes two assumptions. First, it assumes that the circuit used in experiment is sufficiently scrambling that its output probabilities follow the Porter-Thomas distribution. This assumption holds for typical instances of random quantum circuits of sufficient depth. Second, it assumes that the circuit uses enough qubits so that the Porter-Thomas distribution can be approximated with the exponential distribution.

In practice the validity of these assumptions can be confirmed by plotting a histogram of output probabilities and comparing it to the exponential distribution.

The mean of this estimator is the true fidelity f and the variance is

(1 + 2f - f^2) / M

where f is the fidelity and M the number of observations, equal to len(probabilities). This is better than logarithmic XEB (see below) when fidelity is f < 0.32. Since this estimator is unbiased, the variance is equal to the mean squared error of the estimator.

The estimator is intended for use with xeb_fidelity() below.

hilbert_space_dimension Dimension of the Hilbert space on which the channel whose fidelity is being estimated is defined.
probabilities Ideal probabilities of bitstrings observed in experiment.

Estimate of fidelity associated with an experimental realization of a quantum circuit.