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arXiv:1511.00727

Bounding experimental quantum error rates relative to fault-tolerant thresholds

Joel J. Wallman

Published Nov 2, 2015• 25 citations
arXivHTMLPDF

Abstract

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 $\approx 2.45$ 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).

DOI
10.48550/arxiv.1511.00727

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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).
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.
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.
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.
Noise mechanisms in quantum systems can be broadly characterized as either coherent (i.e., unitary) or incoherent. For a given fixed average error rate, coherent noise mechanisms will generally lead to a larger worst-case error than incoherent noise. We show that the coherence of a noise source can be quantified by the unitarity , which we relate to the average change in purity averaged over input pure states. We then show that the unitarity can be efficiently estimated using a protocol based on randomized benchmarking that is efficient and robust to state-preparation and measurement errors. We also show that the unitarity provides a lower bound on the optimal achievable gate infidelity under a given noisy process.
Abstract There is currently a significant need for robust and efficient methods for characterizing quantum devices. While there has been significant progress in this direction, there remains a crucial need to precisely determine the strength and type of errors on individual gate operations, in order to assess and improve control as well as reliably bound the total error in a quantum circuit given some partial information about the errors on the components. In this work, we first provide an optimal bound on the total fidelity of a circuit in terms of component fidelities, which can be efficiently experimentally estimated via randomized benchmarking (RB). We then derive a tighter bound that applies under additional information about the coherence of the error, namely, the unitarity, which can also be estimated via a related experimental protocol. This improved bound smoothly interpolates between the worst-case quadratic and best-case linear scaling for composite error channels. As an application we show how our analysis substantially improves the achievable precision on estimates of the infidelities of individual gates under interleaved RB, enabling greater precision for current experimental methods to assess and tune-up control over quantum gate operations.
While the power of quantum computers is commonly acknowledged to rise exponentially, it is often overlooked that the complexity of quantum noise mechanisms generally grows much faster. In particular, quantifying whether the instructions on a quantum processor are close to being unitary has important consequences concerning error rates, e.g., for the confidence in their estimation, the ability to mitigate them efficiently, or their relation to fault-tolerance thresholds in error correction. However, the complexity of estimating the coherence, or unitarity, of noise generally scales exponentially in system size. Here, we obtain an upper bound on the average unitarity of Pauli noise and develop a protocol allowing us to estimate the average unitarity of operations in a digital quantum device efficiently and feasibly for mid-size quantum systems. We demonstrate our results through both experimental execution on IQM Spark (TM), a 5-qubit superconducting quantum computer, and in simulation with up to 10 qubits, discussing the prospects for extending our technique to arbitrary scales.