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arXiv:1501.04932 · New Journal of Physics

Bounding quantum gate error rate based on reported average fidelity

Yuval R. Sanders, Joel J. Wallman, Barry C. Sanders

Published Dec 21, 2015• 121 citations
arXivHTMLPDF

Abstract

Remarkable experimental advances in quantum computing are exemplified by recent announcements of impressive average gate fidelities exceeding 99.9% for single-qubit gates and 99% for two-qubit gates. Although these high numbers engender optimism that fault-tolerant quantum computing is within reach, the connection of average gate fidelity with fault-tolerance requirements is not direct. Here we use reported average gate fidelity to determine an upper bound on the quantum-gate error rate, which is the appropriate metric for assessing progress towards fault-tolerant quantum computation, and we demonstrate that this bound is asymptotically tight for general noise. Although this bound is unlikely to be saturated by experimental noise, we demonstrate using explicit examples that the bound indicates a realistic deviation between the true error rate and the reported average fidelity. We introduce the Pauli distance as a measure of this deviation, and we show that knowledge of the Pauli distance enables tighter estimates of the error rate of quantum gates.

DOI
10.1088/1367-2630/18/1/012002

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Remarkable experimental advances in quantum computing are exemplified by recent announcements of impressive average gate fidelities exceeding 99.9% for single-qubit gates and 99% for two-qubit gates. Although these high numbers engender optimism that fault-tolerant quantum computing is within reach, the connection of average gate fidelity with fault-tolerance requirements is not direct. Here we use reported average gate fidelity to determine an upper bound on the quantum-gate error rate, which is the appropriate metric for assessing progress towards fault-tolerant quantum computation, and we demonstrate that this bound is asymptotically tight for general noise. Although this bound is unlikely to be saturated by experimental noise, we demonstrate using explicit examples that the bound indicates a realistic deviation between the true error rate and the reported average fidelity. We introduce the Pauli distance as a measure of this deviation, and we show that knowledge of the Pauli distance enables tighter estimates of the error rate of quantum gates.
Gate fidelity -- an average fidelity over all possible input states -- is the workhorse metric for benchmarking quantum gates or circuits, yet fault-tolerant quantum computing ultimately depends on the worst-case behavior, typically quantifiable by so-called the diamond distance. In the low-error regime, the coherent errors can inflate the worst-case error even when the reported gate fidelity is high, making the gate fidelity alone an unreliable proxy for fault-tolerance readiness. To capture the missing information, we introduce a companion observable -- what we dub the fidelity deviation -- that quantifies how strongly the state-dependent fidelities fluctuate across input states. Adopting such fluctuations in assessing the fault-tolerance is physically natural because some input directions are nearly unaffected while others form narrow "valleys" that dominate adversarial circuit behavior. For coherent (unitary) gate errors on two or more qubits, we show that the gate fidelity together with the fidelity deviation constrains the relevant spectral moments of the error unitary, enabling an explicit and tight certificate of the worst-case error. Both quantities are estimated directly from the same randomized input-measurement experiment, without full process tomography. We show that the fidelity and its deviation can provide an economical, operationally meaningful, and accurate standard for assessing the fault tolerance of the engineered quantum gates and circuits.
Rigorously establishing that the error in an experimental quantum operation is beneath the threshold for fault-tolerant quantum computation currently requires considering the worst-case error, which can be orders of magnitude smaller than the average gate infidelities routinely reported in experiments. We show that an improved bound on the worst-case error can be obtained by also considering the recently-introduced unitarity of the noise where the upper and lower bounds differ by a factor of ≈2.45\approx 2.45≈2.45 for unital qubit channels. We prove that the contribution from the nonunital part of any noise map is at most on the order of the average gate infidelity and so is negligible relative to any coherent contribution. We also show that the "average" error rate when measurements are not restricted to an eigenbasis containing the state of the system exhibits the same scaling as the worst-case error, which, for coherent noise, is the square-root of the infidelity. We also obtain improved bounds for the diamond distance when the noise map is known (or approximately known).
Error-free computation is an unattainable ideal, yet our world now contains many computers that appear error-free to their users. That such things are possible is explained by sophisticated theorems that demonstrate the possibility of efficiently reducing computational errors introduced by reasonably well-behaved noise. My thesis is about the problem of determining whether noise in prototype quantum information processors is sufficiently well-behaved for fault-tolerant quantum computing to be possible. My work is divided into two themes. The first theme is the interpretation of average gate fidelity, a quantity that has become the standard performance metric for assessing progress towards fault tolerance. I have elucidated the connection between average gate fidelity and the requirements of fault-tolerant quantum computing by demonstrating the limits of fidelity as a proxy for error rate, the usual metric in fault-tolerance literature. I thereby conclude that information additional to fidelity is required to assess progress towards fault-tolerance. The second theme is the characterization of two-level defect systems, a particularly deleterious kind of noise that can affect superconducting-integrated-circuit-based quantum computing prototypes. I have designed statistical experimental design algorithms that can rigorously assess the influence of these defect systems, and I helped develop a proposal to mitigate their influence. I thereby demonstrate that existing experimental techniques can become much more powerful by employing advanced data collection procedures. My work has immediate implications for current research efforts towards the first working quantum computer. Theoretical work should be directed at assessing noise sources using metrics other than average gate fidelity, and future experimental characterization techniques should become more modular in order to incorporate advanced statistical inference techniques like the ones I develop herein.
We consider how to forecast progress in the domain of quantum computing. For this purpose we collect a dataset of quantum computer systems to date, scored on their physical qubits and gate error rate, and we define an index combining both metrics, the generalized logical qubit. We study the relationship between physical qubits and gate error rate, and tentatively conclude that they are positively correlated (albeit with some room for doubt), indicating a frontier of development that trades-off between them. We also apply a log-linear regression on the metrics to provide a tentative upper bound on how much progress can be expected over time. Within the (generally optimistic) assumptions of our model, including the key assumption that exponential progress in qubit count and gate fidelity will continue, we estimate that that proof-of-concept fault-tolerant computation based on superconductor technology is unlikely (<5% confidence) to be exhibited before 2026, and that quantum devices capable of factoring RSA-2048 are unlikely (<5% confidence) to exist before 2039. It is of course possible that these milestones will in fact be reached earlier, but that this would require faster progress than has yet been seen.
We study how entanglement among the register qubits affects the gate fidelity in the one-way quantum computation if a measurement is inaccurate. We derive an inequality that shows that the mean gate fidelity is upper bounded by a decreasing function of the magnitude of the error of the measurement and the amount of the entanglement between the measured qubit and other register qubits. The consequence of this inequality is that, for a given amount of entanglement, which is theoretically calculated once the algorithm is fixed, we can estimate from this inequality how small the magnitude of the error should be in order not to make the gate fidelity below a threshold, which is specified by a technical requirement in a particular experimental setup or by the threshold theorem of the fault-tolerant quantum computation.
Achieving error rates that meet or exceed the fault-tolerance threshold is a central goal for quantum computing experiments, and measuring these error rates using randomized benchmarking is now routine. However, direct comparison between measured error rates and thresholds is complicated by the fact that benchmarking estimates average error rates while thresholds reflect worst-case behavior when a gate is used as part of a large computation. These two measures of error can differ by orders of magnitude in the regime of interest. Here we facilitate comparison between the experimentally accessible average error rates and the worst-case quantities that arise in current threshold theorems by deriving relations between the two for a variety of physical noise sources. Our results indicate that it is coherent errors that lead to an enormous mismatch between average and worst case, and we quantify how well these errors must be controlled to ensure fair comparison between average error probabilities and fault-tolerance thresholds.