Fair Sampling Error Analysis on NISQ Devices

TL;DR

This paper investigates the fairness of sampling in NISQ quantum devices, revealing how error rates influence the fairness of solutions obtained from quantum circuits, and introduces a new metric for this purpose.

Contribution

It introduces a novel fairness metric based on Pearson's chi-squared test and provides empirical analysis of sampling fairness on IBM Q devices across various error regimes.

Findings

Fair sampling fairness varies with error rates, being higher at low and high error regimes.

Structured errors tend to dominate in medium error regimes, reducing fairness.

Unstructured, random errors are associated with more fair sampling in noisier circuits.

Abstract

We study the status of fair sampling on Noisy Intermediate Scale Quantum (NISQ) devices, in particular the IBM Q family of backends. Using the recently introduced Grover Mixer-QAOA algorithm for discrete optimization, we generate fair sampling circuits to solve six problems of varying difficulty, each with several optimal solutions, which we then run on twenty backends across the IBM Q system. For a given circuit evaluated on a specific set of qubits, we evaluate: how frequently the qubits return an optimal solution to the problem, the fairness with which the qubits sample from all optimal solutions, and the reported hardware error rate of the qubits. To quantify fairness, we define a novel metric based on Pearson's $\chi^2$ test. We find that fairness is relatively high for circuits with small and large error rates, but drops for circuits with medium error rates. This indicates that structured errors dominate in this regime, while unstructured errors, which are random and thus inherently fair, dominate in noisier qubits and longer circuits. Our results show that fairness can be a powerful tool for understanding the intricate web of errors affecting current NISQ hardware.