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arXiv:2110.02290 · Physical Review A

Error metric for non-trace-preserving quantum operations

Yu Bo Shi, Edo Waks A

Published Sep 19, 2023• 3 citations
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Abstract

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.

DOI
10.1103/physreva.108.032609

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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.
The diamond distance can measure the error between real and ideal quantum processes. We can calculate the distance for trace-preserving channels using convex optimization. However, convex optimization cannot solve the distance for non-trace-preserving channels because they are nonlinear. We generalize the diamond distance for non-trace-preserving channels. We first compute the distance for a simple measurement and post-selection channel, then generalize the method to any non-trace-preserving channels. As an example, we analyze the Knill-Laflamme-Milburn (KLM) protocol in linear optical quantum computing with physical imperfections. Our method provides a well-behaved measure for general quantum channels, which can bridge the physical imperfections with the logical errors in quantum circuits. Hence, it has broad applications in analyzing real systems in quantum information processing, such as quantum computers, quantum teleportation, measurement-based quantum computation, etc.
Quantum state comparison, utilizing metrics like fidelity and trace distance, underpins the assessment of quantum networks within quantum information theory. While recent research has expanded theoretical understanding, incorporating error analysis and scalability considerations remains crucial for practical applications. The primary contribution of this letter is to address these gaps by deriving the novel operational trace distance for multi-node networks, establishing a trace distance vs. fidelity benchmark incorporating error bounds, and bridging quantum operations with tensor network analysis. We further explore the application of tensor network tools to quantum networks, offering new analytical avenues. This comprehensive approach provides a robust framework for evaluating quantum network performance under realistic error conditions, facilitating the development of reliable quantum technologies.
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.
Programmable neutral atom arrays show great promise for fault-tolerant quantum computing. A dominant physical error on this platform is qubit leakage and loss, notably decay errors from the Rydberg state during two-qubit gates. Such leakage events are particularly detrimental as they propagate, generating correlated errors that severely degrade the effective error distance of quantum error correction codes. Here, we present a novel approach to address Rydberg decay errors leveraging measurement-based quantum computation. Our scheme strategically exploits the inherent geometric structure of topological cluster states and only uses final leakage detection information to locate propagated errors originating from Rydberg decay. This eliminates the need for complex and atom-species-specific midcircuit leakage detection, offering broader applicability, e.g., to the well-established Rb atom platform. We demonstrate a high error threshold of 3.65% per cz gate for pure Rydberg decay and achieve a favorable error distance d_{e}≈d. Our method compares favorably with state-of-the-art erasure conversion protocols in the subthreshold performance, offering comparable or marginally larger logical error rates while significantly reducing experimental overhead.
In order to place an upper bound on the tolerable amount of photon loss at the detectors, we calculate the error threshold for the linear optics quantum computing proposal by Knill, Laflamme, and Milburn [Nature (London) 409, 46 (2001)] under an error model where photon detectors have efficiency <100<100%<100 but all other components---such as single photon sources, beam splitters, and phase shifters---are perfect and introduce no errors. We make use of the fact that the error model induced by the lossy hardware is that of an erasure channel, i.e., the error locations are always known. Using a method based on a Markov chain description of the error correction procedure, our calculations show that, with the 7 qubit Calderbank--Shor--Steane quantum code, the gate error threshold for fault tolerant quantum computation is bounded below by a value between 1.78% and 11.5% depending on the construction of the entangling gates.