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arXiv:2306.07418

Stochastic errors in quantum instruments

Darian McLaren, Matthew A. Graydon, Joel J. Wallman

Published Jun 12, 2023• 0 citations
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Abstract

Fault-tolerant quantum computation requires non-destructive quantum measurements with classical feed-forward. Many experimental groups are actively working towards implementing such capabilities and so they need to be accurately evaluated. As with unitary channels, an arbitrary imperfect implementation of a quantum instrument is difficult to analyze. In this paper, we define a class of quantum instruments that correspond to stochastic errors and thus are amenable to standard analysis methods. We derive efficiently computable upper- and lower-bounds on the diamond distance between two quantum instruments. Furthermore, we show that, for the special case of uniform stochastic instruments, the diamond distance and the natural generalization of the process infidelity to quantum instruments coincide and are equal to a well-defined probability of an error occurring during the measurement.

DOI
10.48550/arxiv.2306.07418

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Fault-tolerant quantum computation requires non-destructive quantum measurements with classical feed-forward. Many experimental groups are actively working towards implementing such capabilities and so they need to be accurately evaluated. As with unitary channels, an arbitrary imperfect implementation of a quantum instrument is difficult to analyze. In this paper, we define a class of quantum instruments that correspond to stochastic errors and thus are amenable to standard analysis methods. We derive efficiently computable upper- and lower-bounds on the diamond distance between two quantum instruments. Furthermore, we show that, for the special case of uniform stochastic instruments, the diamond distance and the natural generalization of the process infidelity to quantum instruments coincide and are equal to a well-defined probability of an error occurring during the measurement.
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).
We study the problem of measuring errors in non-trace-preserving quantum operations, with a focus on their impact on quantum computing. We propose an error metric that efficiently provides an upper bound on the trace distance between the normalized output states from imperfect and ideal operations, while remaining compatible with the diamond distance. As a demonstration of its application, we apply our metric in the analysis of a lossy beam splitter and a nondeterministic conditional sign-flip gate, two primary non-trace-preserving operations in the Knill-Laflamme-Milburn protocol. We then turn to the leakage errors of neutral-atom quantum computers, finding that these errors scale worse than previously anticipated, implying a more stringent fault-tolerant threshold. We also assess the quantum Zeno gate's error using our metric. In a broader context, we discuss the potential of our metric to analyze general postselected protocols, where it can be employed to study error propagation and estimate thresholds in fault-tolerant quantum computing. The results highlight the critical role of our proposed error metric in understanding and addressing challenges in practical quantum information processing.
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.
Quantum measurements with feed-forward are crucial components of fault-tolerant quantum computers. We show how the error rate of such a measurement can be directly estimated by fitting the probability that successive randomly compiled measurements all return the ideal outcome. Unlike conventional randomized benchmarking experiments and alternative measurement characterization protocols, all the data can be obtained using a single sufficiently large number of successive measurements. We also prove that generalized Pauli fidelities are invariant under randomized compiling and can be combined with the error rate to characterize the underlying errors up to a gauge transformation that introduces an ambiguity between errors happening before or after measurements.
Fault-tolerant quantum computation traditionally incurs substantial resource overhead, with both qubit and time overheads scaling polylogarithmically with the size of the computation. While prior work by Gottesman showed that constant qubit overhead is achievable under stochastic noise using quantum low-density parity-check (QLDPC) codes, it has remained an open question whether similar guarantees hold under more general, non-stochastic noise models. In this work, we address this question by considering a general circuit-level noise model defined via the diamond norm, which captures both stochastic and non-stochastic noise, including coherent and amplitude damping noise. We prove that constant qubit overhead fault-tolerant quantum computation is achievable in this general setting, using QLDPC codes with constant rate and linear minimum distance. To establish our result, we develop a fault-tolerant error correction scheme and a method for implementing logic gates under general circuit noise. These results extend the theoretical foundations of fault-tolerant quantum computation and offer new directions for fault-tolerant architectures under realistic noise models.
In this thesis, we are concerned with the problem of characterizing noise associated with implementations of quantum circuits. We first explore the notion of error rates of quantum circuits and argue that the semantic distinction between process fidelity as “average error rate” and diamond distance as “worst-case error rate” is wrong. To this end, we analyze their worst-case and average analogies, and show they are proportional to their original measures. We show that the non-unital contributions to the diamond distance are negligible and the discrepancy between the process fidelity and diamond distance is primarily due to unitary errors. \n \nWe then turn to a new analysis of cycle benchmarking, a randomized benchmarking-like protocol to estimate the process fidelity of a cycle as engineered by randomized compiling. Using this approach, we prove that gate-dependent noise on the randomized gates is de- scribed by a single perturbation term that decays rapidly as long as the implementation is close to a representation. We also comment on how the protocol can be extended to qudits, what the cycle benchmarking decay actually measures, and how our analysis is amenable to the Fourier analysis of randomized benchmarking. We end with a discussion of how the gate-dependent cycle benchmarking process fidelity relates to gate-dependent randomized compiling.