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arXiv:1703.09835 · Quantum

Randomized benchmarking with gate-dependent noise

Joel J. Wallman

Published Jan 29, 2018• 102 citations
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Abstract

We analyze randomized benchmarking for arbitrary gate-dependent noise and prove that the exact impact of gate-dependent noise can be described by a single perturbation term that decays exponentially with the sequence length. That is, the exact behavior of randomized benchmarking under general gate-dependent noise converges exponentially to a true exponential decay of exactly the same form as that predicted by previous analysis for gate-independent noise. Moreover, we show that the operational meaning of the decay parameter for gate-dependent noise is essentially unchanged, that is, we show that it quantifies the average fidelity of the noise between ideal gates. We numerically demonstrate that our analysis is valid for strongly gate-dependent noise models. We also show why alternative analyses do not provide a rigorous justification for the empirical success of randomized benchmarking with gate-dependent noise.

DOI
10.22331/q-2018-01-29-47

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We analyze randomized benchmarking for arbitrary gate-dependent noise and prove that the exact impact of gate-dependent noise can be described by a single perturbation term that decays exponentially with the sequence length. That is, the exact behavior of randomized benchmarking under general gate-dependent noise converges exponentially to a true exponential decay of exactly the same form as that predicted by previous analysis for gate-independent noise. Moreover, we show that the operational meaning of the decay parameter for gate-dependent noise is essentially unchanged, that is, we show that it quantifies the average fidelity of the noise between ideal gates. We numerically demonstrate that our analysis is valid for strongly gate-dependent noise models. We also show why alternative analyses do not provide a rigorous justification for the empirical success of randomized benchmarking with gate-dependent noise.
Randomized benchmarking (RB) has emerged as a popular and easy-to-implement experimental technique for gauging the quality of gate operations in quantum computing devices. A typical randomized benchmarking procedure identifies the exponential decay in the fidelity as the benchmarking sequence of gates increases in length, and the decay rate is used to estimate the fidelity of the gate. Our current theory of RB, based on the assumption of time-independent or static noise in the gates, predicts an exponential decay in the fidelity. Observed deviations from the standard exponential decay are usually attributed to some amount of time correlations in the noise, though the precise mechanisms for deviation have yet to be fully explored. In this paper, we examine this question of randomized benchmarking for time-correlated noise---specifically for time-correlated dephasing noise for exact solvability---and elucidate the circumstances in which a deviation from exponential decay can be expected, as well as point out situations where exponential decay prevails despite the presence of time correlations.
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.
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 refers to a collection of protocols that in the past decade have become central methods for characterizing quantum gates.These protocols aim at efficiently estimating the quality of a set of quantum gates in a way that is resistant to state preparation and measurement errors.Over the years many versions have been developed, however a comprehensive theoretical treatment of randomized benchmarking has been missing.In this work, we develop a rigorous framework of randomized benchmarking general enough to encompass virtually all known protocols as well as novel, more flexible extensions.Overcoming previous limitations on error models and gate sets, this framework allows us, for the first time, to formulate realistic conditions under which we can rigorously guarantee that the output of any randomized benchmarking experiment is well described by a linear combination of matrix exponential decays.We complement this with a detailed analysis of the fitting problem associated with randomized benchmarking data.We introduce modern signal processing techniques to randomized benchmarking, prove analytical sample complexity bounds, and numerically evaluate performance and limitations.In order to reduce the resource demands of this fitting problem, we introduce novel, scalable postprocessing techniques to isolate exponential decays, significantly improving the practical feasibility of a large set of randomized benchmarking protocols.These postprocessing techniques overcome shortcomings in efficiency of several previously proposed methods such as character benchmarking and linear-cross entropy benchmarking.Finally, we discuss, in full generality, how and when randomized benchmarking decay rates can be used to infer quality measures like the average fidelity.On the technical side, our work substantially extends the recently developed Fourier-theoretic perspective on randomized benchmarking by making use of the perturbation theory of invariant subspaces, as well as ideas from signal processing.
We analyze the impact of non-Markovian classical noise on single-qubit randomized benchmarking experiments, in a manner that explicitly models the realization of each gate via realistic finite-duration pulses. Our new framework exploits the random nature of each gate sequence to derive expressions for the full survival probability decay curve which are nonperturbative in the noise strength. In the presence of non-Markovian noise, our approach shows that the decay curve can exhibit a strong dependence on the gate implementation method, with regimes of both exponential and power law decays. We discuss how these effects can complicate the interpretation of a randomized benchmarking experiment, but also how to leverage them to probe non-Markovianity.
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.