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arXiv:2411.18715 · Physical review. B./Physical review. B

Model validation and error attribution for a drifting qubit

Malick A. Gaye, Dylan Albrecht, Steve M. Young, Tameem Albash, N. Tobias Jacobson

Published Mar 11, 2025• 1 citation
arXivHTMLPDF

Abstract

Qubit performance is often reported in terms of a variety of single-value metrics, each providing a facet of the underlying noise mechanism limiting performance. However, the value of these metrics may drift over long timescales, and reporting a single number for qubit performance fails to account for the low-frequency noise processes that give rise to this drift. In this work, we demonstrate how we can use the distribution of these values to validate or invalidate candidate noise models. We focus on the case of randomized benchmarking (RB), where typically a single error rate is reported but this error rate can drift over time when multiple passes of RB are performed. We show that using a statistical test as simple as the Kolmogorov-Smirnov statistic on the distribution of RB error rates can be used to rule out noise models, assuming the experiment is performed over a long enough time interval to capture relevant low frequency noise. With confidence in a noise model, we show how care must be exercised when performing error attribution using the distribution of drifting RB error rate.

DOI
10.1103/physrevb.111.115303

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Qubit performance is often reported in terms of a variety of single-value metrics, each providing a facet of the underlying noise mechanism limiting performance. However, the value of these metrics may drift over long timescales, and reporting a single number for qubit performance fails to account for the low-frequency noise processes that give rise to this drift. In this work, we demonstrate how we can use the distribution of these values to validate or invalidate candidate noise models. We focus on the case of randomized benchmarking (RB), where typically a single error rate is reported but this error rate can drift over time when multiple passes of RB are performed. We show that using a statistical test as simple as the Kolmogorov-Smirnov statistic on the distribution of RB error rates can be used to rule out noise models, assuming the experiment is performed over a long enough time interval to capture relevant low frequency noise. With confidence in a noise model, we show how care must be exercised when performing error attribution using the distribution of drifting RB error rate.
Among the most popular and well-studied quantum characterization, verification, and validation techniques is randomized benchmarking (RB), an important statistical tool used to characterize the performance of physical logic operations useful in quantum information processing. In this work we provide a detailed mathematical treatment of the effect of temporal noise correlations on the outcomes of RB protocols. We provide a fully analytic framework capturing the accumulation of error in RB expressed in terms of a three-dimensional random walk in ``Pauli space.'' Using this framework we derive the probability density function describing RB outcomes (averaged over noise) for both Markovian and correlated errors, which we show is generally described by a $\mathrm{\ensuremath{\Gamma}}$ distribution with shape and scale parameters depending on the correlation structure. Long temporal correlations impart large nonvanishing variance and skew in the distribution towards high-fidelity outcomes---consistent with existing experimental data---highlighting potential finite-sampling pitfalls and the divergence of the mean RB outcome from worst-case errors in the presence of noise correlations. We use the filter-transfer function formalism to reveal the underlying reason for these differences in terms of effective coherent averaging of correlated errors in certain random sequences. We conclude by commenting on the impact of these calculations on the utility of single-metric approaches to quantum characterization, verification, and validation.
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.
In this paper, we analyze the performance of randomized benchmarking protocols on gate sets under a variety of realistic error models that include systematic rotations, amplitude damping, leakage to higher levels, and 1/f1/f1/f noise. We find that, in almost all cases, benchmarking provides better than a factor-of-2 estimate of average error rate, suggesting that randomized benchmarking protocols are a valuable tool for verification and validation of quantum operations. In addition, we derive models for fidelity decay curves under certain types of non-Markovian noise models such as 1/f1/f1/f and leakage errors. We also show that, provided the standard error of the fidelity measurements is small, only a small number of trials are required for high-confidence estimation of gate errors.
Randomized benchmarking (RB) is a widely used strategy to assess the quality of available quantum gates in a computational context. RB involves applying known random sequences of gates to an initial state and using the statistics of a final measurement step to determine an effective depolarizing error per step of the sequence, which is a metric of gate quality. Here we investigate the advantages of fully randomized benchmarking, where a new random sequence is drawn for each experimental trial. The advantages of full randomization include smaller confidence intervals on the inferred step error, the ability to use maximum likelihood analysis without heuristics, straightforward optimization of the sequence lengths, and the ability to model and measure behaviors that go beyond the typical assumption of time-independent error rates. We discuss models of time-dependent or non-Markovian errors that generalize the basic RB model of a single exponential decay of the success probability. For any of these models, we implement a concrete protocol to minimize the uncertainty of the estimated parameters given a fixed time constraint on the complete experiment, and we implement a maximum likelihood analysis. We consider several previously published experiments and determine the potential for improvements with optimized full randomization. We experimentally observe such improvements in Clifford randomized benchmarking experiments on a single trapped ion qubit at the National Institute of Standards and Technology (NIST). For an experiment with uniform lengths and intentionally repeated sequences the step error was 2.42−0.22+0.30×10−52.42^{+0.30}_{-0.22}\times 10^{-5}2.42−0.22+0.30​×10−5, and for an optimized fully randomized experiment of the same total duration the step error was 2.57−0.06+0.07×10−52.57^{+0.07}_{-0.06}\times 10^{-5}2.57−0.06+0.07​×10−5. We find a substantial decrease in the uncertainty of the step error as a result of optimized fully randomized benchmarking.
Randomized benchmarking (RB) is an efficient and robust method to characterize gate errors in quantum circuits. Averaging over random sequences of gates leads to estimates of gate errors in terms of the average fidelity. These estimates are isolated from the state preparation and measurement errors that plague other methods such as channel tomography and direct fidelity estimation. A decisive factor in the feasibility of randomized benchmarking is the number of sampled sequences required to obtain rigorous confidence intervals. Previous bounds were either prohibitively loose or required the number of sampled sequences to scale exponentially with the number of qubits in order to obtain a fixed confidence interval at a fixed error rate. Here, we show that, with a small adaptation to the randomized benchmarking procedure, the number of sampled sequences required for a fixed confidence interval is dramatically smaller than could previously be justified. In particular, we show that the number of sampled sequences required is essentially independent of the number of qubits and scales favorably with the average error rate of the system under investigation. We also investigate the fitting procedure inherent to randomized benchmarking in light of our results and find that standard methods such as ordinary least squares optimization can give misleading results. We therefore recommend moving to more sophisticated fitting methods such as iteratively reweighted least squares optimization. Our results bring rigorous randomized benchmarking on systems with many qubits into the realm of experimental feasibility.
Advancing quantum information processors and building fault-tolerant architectures rely on the ability to accurately characterize the noise sources and suppress their impact on quantum devices. In practice, noise often drifts over time, whereas conventional noise characterization and decoding methods typically assume stationarity or provide only a time-average behavior of the noise. This treatment can result in suboptimal decoding performance. In this work, we present a rigorous analytical framework to capture time-dependent Pauli noise, by exploiting the syndrome statistics of quantum error correction experiments. We propose a sliding-window estimation method which allows us to recover the frequency components of the noise, by using optimal window sizes that we derive analytically. We prove the noise-filtering behavior of sliding windows, linking window size to spectral cutoff frequencies, and provide an iterative algorithm that captures multiple drift frequencies. We further introduce an overlapping window algorithm that enables us to capture rapid multi-frequency noise drifts in a single-pass fashion. Simulations for both phenomenological and circuit-level noise models validate our framework, demonstrating robust tracking of multi-frequency drift. The logical error rate obtained from our estimated models consistently align with the ground-truth logical error rate, and we find suppression of logical errors compared to static error models. Our window-based estimation methods and adaptive decoding offer new insights into noise spectroscopy and decoder optimization under drift using only syndrome data.