What Randomized Benchmarking Actually Measures

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.

Experimental quantum verification in the presence of temporally correlated noise

“…The asymmetry we observe between the manifestation of correlatedσ x =σ y andσ z errorsensitivity has previously only been reported in the context of RB. 23 We have shown explicitly how the low diamond-distance estimates under this kind of noise are related to the gauge optimisation performed as part of the protocol; limiting the gauge freedom by extending the gate set under application of an identical error process dramatically changed the estimated diamond distance of the very same gates in numerical simulations. This highlights that estimates are always reported up to an implicit gauge degree of freedom, making absolute comparisons of diamond norms challenging.…”

“…The relevance of this gauge freedom on RB-derived estimates of gate performance was highlighted recently in. 23 To illustrate how gauge freedom affects the results, we separately calculate the diamond distance with and without gauge optimising our analytic gate set G err using routines included in the pyGSTi toolkit.…”

Gate-error analysis in simulations of quantum computers with transmon qubits

“…This can have three reasons: (i) the procedure of measuring the fidelities (i.e., randomized benchmarking) produces numbers that overestimate the gate performance (cf. [33,45,46]), implying that the actual gate implementations are worse; (ii) the actual gates are good but the discrepancy is due to another process (such as the measurement) that is not yet included in the simulation model; or (iii) other unknown factors not included in the quantum-theoretical description of the experiments play a destructive role.…”

Relations between single and repeated qubit gates: coherent error amplification for high-fidelity quantum-gate tomography

“…This estimate remains valid in the vicinity of the other peaks at ξ=(2k+1)α/(2n) (k=1, 2, K) upon the replacement α → (2k+1) α in equation (29). The single-pass transition probability p can be found by the Figure 5.…”

“…However, it delivers an error value which is not exactly the error of a particular gate but an average error over several gates (π x , π y , π/2), which may have different errors. Recently, there has been some discussion about what randomized benchmarking actually measures [29,30].…”