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arXiv:2610.31500

The Impact of Markovian Errors on Random Circuits

Anthony M. Polloreno, Timothy Proctor, Robin Blume-Kohout

quant-phSubmitted Oct 22, 2026• Announced Oct 23, 2026• 0 citations
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Abstract

Randomized benchmarking (RB) returns a single decay parameter or gate error rate, but when the noise is gate-dependent that parameter is notoriously difficult to interpret. We study RB and related random-circuit benchmarks under small gate-local Markovian noise by factorizing each noisy gate into its ideal action followed by the exponential of a small error generator and then perturbing the RB $\mathscr{L}$ matrix, which models the RB decay. For any low-error gate set that satisfies the requirements of RB (it is a sequence-asymptotic unitary 2-design), we show that Pauli-stochastic error rates contribute linearly to the RB decay parameter, while Hamiltonian error rates contribute through a gauge-invariant quadratic form. Pauli-correlation and active generators are absent from the linear term and enter only at higher order. For the single-qubit native gate set $\{X_{π/2},Y_{π/2}\}$ we derive this quadratic form explicitly in terms of three first-order gauge-invariant (FOGI) coordinates: the two intrinsic over-rotations and one relational axis-tilt error. The resulting formula explains the dependence of DRB on the sampling distribution and the compilation dependence of CRB. These results turn RB decay rates into interpretable functions of experimentally meaningful elementary error rates.

Comments
15 pages, 3 figures

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Randomized benchmarking (RB) returns a single decay parameter or gate error rate, but when the noise is gate-dependent that parameter is notoriously difficult to interpret. We study RB and related random-circuit benchmarks under small gate-local Markovian noise by factorizing each noisy gate into its ideal action followed by the exponential of a small error generator and then perturbing the RB L\mathscr{L}L matrix, which models the RB decay. For any low-error gate set that satisfies the requirements of RB (it is a sequence-asymptotic unitary 2-design), we show that Pauli-stochastic error rates contribute linearly to the RB decay parameter, while Hamiltonian error rates contribute through a gauge-invariant quadratic form. Pauli-correlation and active generators are absent from the linear term and enter only at higher order. For the single-qubit native gate set {Xπ/2,Yπ/2}\{X_{π/2},Y_{π/2}\}{Xπ/2​,Yπ/2​} we derive this quadratic form explicitly in terms of three first-order gauge-invariant (FOGI) coordinates: the two intrinsic over-rotations and one relational axis-tilt error. The resulting formula explains the dependence of DRB on the sampling distribution and the compilation dependence of CRB. These results turn RB decay rates into interpretable functions of experimentally meaningful elementary error rates.
Randomized benchmarking (RB) protocols have become an essential tool for providing a meaningful partial characterization of experimental quantum operations. While the RB decay rate is known to enable estimates of the average fidelity of those operations under gate-independent Markovian noise, under gate-dependent noise this rate is more difficult to interpret rigorously. In this paper, we prove that single-qubit RB decay parameter p coincides with the decay parameter of the gate-set circuit fidelity , a novel figure of merit which characterizes the expected average fidelity over arbitrary circuits of operations from the gate-set. We also prove that, in the limit of high-fidelity single-qubit experiments, the possible alarming disconnect between the average gate fidelity and RB experimental results is simply explained by a basis mismatch between the gates and the state-preparation and measurement procedures, that is, to a unitary degree of freedom in labeling the Pauli matrices. Based on numerical evidence and physically motivated arguments, we conjecture that these results also hold for higher dimensions.
Randomized benchmarking (RB) protocols have become an essential tool for providing a meaningful partial characterization of experimental quantum operations. While the RB decay rate is known to enable estimates of the average fidelity of those operations under gate-independent Markovian noise, under gate-dependent noise this rate is more difficult to interpret rigorously. In this paper, we prove that single-qubit RB experiments provide direct estimates of the gateset circuit fidelity, a novel figure of merit which characterizes the expected average fidelity over arbitrary circuits of operations from the gateset. We also prove that, in the limit of high-fidelity single-qubit experiments, the difficulty introduced by gate-dependence reduces to a basis mismatch between the gates and the state-preparation and measurement procedures, that is, to a unitary degree of freedom in labeling the Pauli matrices. Based on numerical evidence and physically motivated arguments, we conjecture that these result hold for higher dimensions.
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.
Randomized benchmarking (RB) protocols are widely used to measure an average error rate for a set of quantum logic gates. However, the standard version of RB is limited because it only benchmarks a processor's native gates indirectly, by using them in composite n -qubit Clifford gates. Standard RB's reliance on n -qubit Clifford gates restricts it to the few-qubit regime, because the fidelity of a typical composite n -qubit Clifford gate decreases rapidly with increasing n . Furthermore, although standard RB is often used to infer the error rate of native gates, by rescaling standard RB's error per Clifford to an error per native gate, this is an unreliable extrapolation. Direct RB is a method that addresses these limitations of standard RB, by directly benchmarking a customizable gate set, such as a processor's native gates. Here we provide a detailed introduction to direct RB, we discuss how to design direct RB experiments, and we present two complementary theories for direct RB. The first of these theories uses the concept of error propagation or scrambling in random circuits to show that direct RB is reliable for gates that experience stochastic Pauli errors. We prove that the direct RB decay is a single exponential, and that the decay rate is equal to the average infidelity of the benchmarked gates, under broad circumstances. This theory shows that group twirling is not required for reliable RB. Our second theory proves that direct RB is reliable for gates that experience general gate-dependent Markovian errors, using similar techniques to contemporary theories for standard RB. Our two theories for direct RB have complementary regimes of applicability, and they provide complementary perspectives on why direct RB works. Together these theories provide comprehensive guarantees on the reliability of direct RB.
In its many variants, randomized benchmarking (RB) is a broadly used technique for assessing the quality of gate implementations on quantum computers. A detailed theoretical understanding and general guarantees exist for the functioning and interpretation of RB protocols if the gates under scrutiny are drawn uniformly at random from a compact group. In contrast, many practically attractive and scalable RB protocols implement random quantum circuits with local gates randomly drawn from some gate-set. Despite their abundance in practice, for those non-uniform RB protocols, general guarantees for gates from arbitrary compact groups under experimentally plausible assumptions are missing. In this work, we derive such guarantees for a large class of RB protocols for random circuits that we refer to as filtered RB. Prominent examples include linear cross-entropy benchmarking, character benchmarking, Pauli-noise tomography and variants of simultaneous RB. Building upon recent results for random circuits, we show that many relevant filtered RB schemes can be realized with random quantum circuits in linear depth, and we provide explicit small constants for common instances. We further derive general sample complexity bounds for filtered RB. We show filtered RB to be sample-efficient for several relevant groups, including protocols addressing higher-order cross-talk. Our theory for non-uniform filtered RB is, in principle, flexible enough to design new protocols for non-universal and analog quantum simulators.
We show that the Randomized Benchmarking (RB) protocol is a convolution amenable to Fourier space analysis. By adopting the mathematical framework of Fourier transforms of matrix-valued functions on groups established in recent work from Gowers and Hatami \cite{GH15}, we provide an alternative proof of Wallman's \cite{Wallman2018} and Proctor's \cite{Proctor17} bounds on the effect of gate-dependent noise on randomized benchmarking. We show explicitly that as long as our faulty gate-set is close to the targeted representation of the Clifford group, an RB sequence is described by the exponential decay of a process that has exactly two eigenvalues close to one and the rest close to zero. This framework also allows us to construct a gauge in which the average gate-set error is a depolarizing channel parameterized by the RB decay rates, as well as a gauge which maximizes the fidelity with respect to the ideal gate-set.