Benchmarking quantum logic operations relative to thresholds for fault tolerance

Abstract

Contemporary methods for benchmarking noisy quantum processors typically measure average error rates or process infidelities. However, thresholds for fault-tolerant quantum error correction are given in terms of worst-case error rates—defined via the diamond norm—which can differ from average error rates by orders of magnitude. One method for resolving this discrepancy is to randomize the physical implementation of quantum gates, using techniques like randomized compiling (RC). In this work, we use gate set tomography to perform precision characterization of a set of two-qubit logic gates to study RC on a superconducting quantum processor. We find that, under RC, gate errors are accurately described by a stochastic Pauli noise model without coherent errors, and that spatially correlated coherent errors and non-Markovian errors are strongly suppressed. We further show that the average and worst-case error rates are equal for randomly compiled gates, and measure a maximum worst-case error of 0.0197(3) for our gate set. Our results show that randomized benchmarks are a viable route to both verifying that a quantum processor’s error rates are below a fault-tolerance threshold, and to bounding the failure rates of near-term algorithms, if—and only if—gates are implemented via randomization methods which tailor noise.

Operational Markovianization in randomized benchmarking

“…The reason behind this result, and for the apparent contradiction with those of [35,54], is that average gate-fidelity as a figure of merit only takes into account the probability of no error happening on S and not all the other error terms that Pauli-twirling eliminates. That is, average gate-fidelity by definition is proportional only to the zero th (identity) element of the χ-matrix; explicitly, see equations (D11) and (D12) in appendix D.1.…”

“…While the Markovianizing effect of DD in RB is somehow expected, a prominent error-suppression technique that has recently been shown to suppress non-Markovian noise in a statistically significant way [35,54] is RC [40,55]. RC can be understood as the operational way of tailoring arbitrary Markovian noise quantum channels into Pauli channels, which mathematically corresponds to a mapping known as Pauli-twirling.…”

“…Pauli-twirling. While for a single-qubit this can be done exactly, in general, the tailoring of noise into Pauli noise by RC can be quantified to occur to a large percentage with a small number of samples [55], generally scaling as an inverse fraction of the number of samples [54]. The only caveats to keep in mind is that this sampling overhead of RC would compound with that of RB [56] and that the Pauli gates could themselves be significantly noisy.…”

“…Other Markovianizing strategies different from DD could also be conceivable. In particular, techniques tailoring noise into a so-called Pauli channel have been shown to reduce the impact of non-Markovian noise [35,39,54] in a statistically significant way.…”