  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 ten different backends
available on 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 provide a simple, intuitive means of quantifying fairness in quantum
circuits, and show that reducing structured errors is necessary to improve fair
sampling on NISQ hardware.
