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arXiv:1504.04057 · Physical Review Letters

Optimizing the Frequency of Quantum Error Correction

Ali Abu-Nada, Ben Fortescue, Mark Byrd

Published Nov 9, 2017• 2 citations
arXivHTMLPDF

Abstract

A common assumption is that one applies a fault-tolerant quantum error correction (FTQEC) after every gate during quantum computing. However, it is known that this is not always optimal, since the FTQEC procedure itself can introduce errors. Here we vary the number of logical gates between FTQEC operations given that a failure of a postselection condition may cause the FTQEC to be skipped. We derive an expression for the logical error rate as a function of the error-correction frequency and find the optimal frequency for the application of the FTQEC. Furthermore, we show that this is relatively insensitive to the postselection failure probability for a large range of such probabilities. We provide an example of the application of the analytic expression to the ⟦7,1,3⟧ Steane code and data derived from a Monte Carlo simulation.

DOI
10.1103/physrevlett.119.190502

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A common assumption is that one applies a fault-tolerant quantum error correction (FTQEC) after every gate during quantum computing. However, it is known that this is not always optimal, since the FTQEC procedure itself can introduce errors. Here we vary the number of logical gates between FTQEC operations given that a failure of a postselection condition may cause the FTQEC to be skipped. We derive an expression for the logical error rate as a function of the error-correction frequency and find the optimal frequency for the application of the FTQEC. Furthermore, we show that this is relatively insensitive to the postselection failure probability for a large range of such probabilities. We provide an example of the application of the analytic expression to the ⟦7,1,3⟧ Steane code and data derived from a Monte Carlo simulation.
Quantum error correction (QEC) is essential for realizing scalable quantum computation. However, when evaluating its benefits, most analyses assume idealized components, overlooking the imperfections inherent in realistic fault-tolerant (FT) implementations. In this paper, we investigate the performance of QEC schemes taking into account that quantum gates and measurements are themselves error-prone. We derive bounds for the failure probability of Shor-style FT-QEC schemes using limited structural information, such as the number of flag qubits and quantum gates. Our analysis separates and quantifies two key contributors to the failure rate: decoding errors and residual errors arising from circuit-level faults. The derived bounds highlight fundamental limitations in Shor-style FT-QEC performance and quantify how circuit imperfections degrade error correction capabilities, under the assumption of depolarizing noise.
Fault-tolerant quantum error correction (FTQEC) is pivotal for ensuring the reliability and scalability of quantum computers, which harness quantum mechanics principles. This paper presents a comprehensive analysis of FTQEC methodologies, integrating theoretical investigations, experimental validations, error correction capabilities, and simulation comparisons. Theoretical frameworks reveal high error correction capabilities, with success rates ranging from 0.88 to 0.95 and significant error rate reductions of up to 98%. Experimental validations across diverse quantum hardware platforms demonstrate promising results, with high success rates and notable error correction efficiency, albeit challenges in scalability. Analysis of error correction capabilities highlights robustness against errors and preservation of qubits integrity during error correction procedures. Simulation comparisons further validate the efficacy of FTQEC protocols, showing significant improvements in error rate reduction and gate fidelity. However, scalability and resource overhead remain concerns, necessitating further research for efficient and scalable FTQEC protocols. Overall, this study contributes to the understanding and advancement of fault-tolerant quantum error correction, crucial for realizing the transformative potential of quantum computing technology.
Scaling up quantum algorithms to tackle high-impact problems in science and industry requires quantum error correction and fault tolerance. While progress has been made in experimentally realizing error-corrected primitives, the end-to-end execution of logical quantum algorithms using only fault-tolerant (FT) components has remained out of reach. We demonstrate the FT and error-corrected execution of two quantum algorithms, the Quantum Approximate Optimization Algorithm (QAOA) and the Harrow-Hassidim-Lloyd (HHL) algorithm applied to the Poisson equation, on Quantinuum H2 and Helios trapped-ion quantum processors using the [[7,1,3]][[7,1,3]][[7,1,3]] Steane code. For QAOA circuits on 5 and 6 logical qubits, we show performance improvements from increasing the number of QAOA layers and the number of TTT gates used to approximate logical rotations, despite increased physical circuit complexity. We further show that QAOA circuits with up to 8 logical qubits and 9 logical TTT gates perform similarly to unencoded circuits. For the largest QAOA circuits we run, with 12 logical (97 physical) qubits and 2132 physical two-qubit gates, we still observe better-than-random performance. Finally, we show that adding active QEC cycles and increasing the repeat-until-success limit of state preparation subroutines can improve the performance of a quantum algorithm, thereby demonstrating critical capabilities of scalable FT quantum computation. Our results are enabled by an FT logical TTT gate implementation with an infidelity of ∼2.6(4)×10−3\sim 2.6(4)\times10^{-3}∼2.6(4)×10−3 and dynamic circuits with measurement-dependent feedback. Our work demonstrates near-break-even performance of complex, error-corrected algorithmic quantum circuits using only FT components.
In general, fault-tolerant quantum error correction (FTQEC) procedures are designed to detect, correct, and be fault-tolerant against errors occurring within the qubit subspace. But in some qubit implementations, additional "leakage" errors can occur in which the system leaves this subspace, and standard FTQEC procedures may not be fault-tolerant against such errors. Generic methods for achieving fault-tolerance against leakage are costly in terms of resources. In this paper we demonstrate that for a leakage model common to many photonic gate implementations, FTQEC can be implemented with far fewer additional operations than in the generic case.
Encoding information redundantly using quantum error-correcting (QEC) codes allows one to overcome the inherent sensitivity to noise in quantum computers to ultimately achieve large-scale quantum computation. The Steane QEC method involves preparing an auxiliary logical qubit of the same QEC code as used for the data register. The data and auxiliary registers are then coupled with a logical controlled- () gate, enabling a measurement of the auxiliary register to reveal the error syndrome. This study presents the implementation of multiple rounds of fault-tolerant (FT) Steane QEC on a trapped-ion quantum computer. Various QEC codes are employed and the results are compared to a previous experimental approach utilizing flag qubits. Our experimental findings show improved logical fidelities for Steane QEC and accompanying numerical simulations indicate an even larger performance advantage for quantum processors limited by entangling-gate errors. This establishes experimental Steane QEC as a competitive paradigm for FT quantum computing. Published by the American Physical Society 2024
Fault-tolerant quantum computers rely on Quantum Error-Correcting Codes (QECCs) to protect information from noise. However, no single error-correcting code supports a fully transversal and therefore fault-tolerant implementation of all gates required for universal quantum computation. Code switching addresses this limitation by moving quantum information between different codes that, together, support a universal gate set. Unfortunately, each switch is costly-adding time and space overhead and increasing the logical error rate. Minimizing the number of switching operations is, therefore, essential for quantum computations using code switching. In this work, we study the problem of minimizing the number of code switches required to run a given quantum circuit. We show that this problem can be solved efficiently in polynomial time by reducing it to a minimum-cut instance on a graph derived from the circuit. Our formulation is flexible and can incorporate additional considerations, such as reducing depth overhead by preferring switches during idle periods or biasing the compilation to favor one code over another. To the best of our knowledge, this is the first automated approach for compiling and optimizing code-switching-based quantum computations at the logical level.