What Randomized Benchmarking Actually Measures

TLDR

Randomized benchmarking estimates a quantum gate set’s error rate by applying random circuits that ideally cancel out, with the exponential decay of survival probabilities yielding a single metric r that was thought to reflect the average gate infidelity over Clifford gates. The authors aim to develop accurate theories of the RB decay for all small process‑matrix errors and demonstrate that the decay curve remains a simple exponential. They construct new theoretical models that account for arbitrary small errors described by process matrices, ensuring the RB decay is exponential across all such errors. Their analysis shows that r is not a well‑defined property of a physical gate set, depends on the chosen representations, can differ by orders of magnitude from the commonly computed variant, and does not correspond to the infidelity of any completely positive representation of the imperfect gates.

Abstract

Randomized benchmarking (RB) is widely used to measure an error rate of a set of quantum gates, by performing random circuits that would do nothing if the gates were perfect. In the limit of no finite-sampling error, the exponential decay rate of the observable survival probabilities, versus circuit length, yields a single error metric r. For Clifford gates with arbitrary small errors described by process matrices, r was believed to reliably correspond to the mean, over all Clifford gates, of the average gate infidelity between the imperfect gates and their ideal counterparts. We show that this quantity is not a well-defined property of a physical gate set. It depends on the representations used for the imperfect and ideal gates, and the variant typically computed in the literature can differ from r by orders of magnitude. We present new theories of the RB decay that are accurate for all small errors describable by process matrices, and show that the RB decay curve is a simple exponential for all such errors. These theories allow explicit computation of the error rate that RB measures (r), but as far as we can tell it does not correspond to the infidelity of a physically allowed (completely positive) representation of the imperfect gates.