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arXiv:1912.09635 · IEEE Transactions on Quantum Engineering

Decoding Quantum Error Correction Codes With Local Variation

Michael Hanks, William J. Munro, Kae Nemoto

Published Jan 1, 2020• 1 citation
arXivHTMLPDF

Abstract

In this article, we investigate the role of local information in the decoding of the repetition and surface error correction codes for the protection of quantum states. Our key result is an improvement in resource efficiency when local information is taken into account during the decoding process: the code distance associated with a given logical error rate is reduced with a magnitude depending on the proximity of the physical error rate to the accuracy threshold of the code. We also briefly discuss an averaged approach with local information for table lookup and localized decoding schemes, an expected breakdown of these effects for large-scale systems, and the importance of this resource reduction in the near term.

DOI
10.1109/tqe.2020.2967890

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In this article, we investigate the role of local information in the decoding of the repetition and surface error correction codes for the protection of quantum states. Our key result is an improvement in resource efficiency when local information is taken into account during the decoding process: the code distance associated with a given logical error rate is reduced with a magnitude depending on the proximity of the physical error rate to the accuracy threshold of the code. We also briefly discuss an averaged approach with local information for table lookup and localized decoding schemes, an expected breakdown of these effects for large-scale systems, and the importance of this resource reduction in the near term.
Abstract Imperfect measurements are a prevalent source of error across quantum computing platforms, significantly degrading the logical error rates achievable on current hardware. To mitigate this issue, rich measurement data referred to as soft information has been proposed to efficiently identify and correct measurement errors as they occur. In this work, we model soft information decoding across a variety of physical qubit platforms and decoders and showcase how soft information can make error correction viable at lower code distances and higher physical error rates than is otherwise possible. We simulate the effects of soft information decoding on quantum memories for surface codes and bivariate bicycle codes, and evaluate the error suppression performance of soft decoders against traditional decoders. Our simulations show that soft information decoding on near-term devices can provide up to 11% higher error suppression on superconducting qubits and up to 20% stronger error suppression on neutral atom qubits. These accuracy gains correspond to 13% and 33% reductions in the physical qubit footprint of superconducting and neutral atom devices respectively when operating at a logical error rate of 10 −6 , showcasing that soft information is a powerful tool for reducing the cost and complexity of large-scale fault-tolerant quantum computers.
This chapter provides an overview of how to perform quantum error correction using the surface code, which is the most well-studied quantum error correcting code for practical quantum computation. We provide formulas for the code distance—which determines the resource overhead when using the surface code—as a function of the desired logical error rate and underlying physical error rate. We discuss several decoders for the surface code and the possibility of experiencing the backlog problem if the decoder is too slow.
Quantum error correcting codes can enable large quantum computations provided physical error rates are sufficiently low. We combine post-selection with surface code error correction through the use of exclusive decoders, which abort on decoding instances that are deemed too difficult. For the most discriminating of exclusive decoders, we demonstrate a threshold of 50% under depolarizing noise (or 32(1)% for the fault-tolerant case), and up to a quadratic improvement in logical failure rates below threshold. Furthermore, with a modest exclusion criterion, we identify a regime at low error rates where the exclusion rate decays with code distance, providing a pathway for scalable and time-efficient quantum computing with post-selection. Our exclusive decoder applied to magic state distillation yields a 75% reduction in the number of physical qubits, and a 60% reduction in the total spacetime volume, including accounting for repetitions. Other applications include error mitigation, and in concatenated schemes. Quantum error correction produces an enormous amount of data about the quantum system, including information about whether an uncorrectable error is likely. In this work the authors analyse a new decoder that can abort when decoding is deemed too difficult, yielding improved performance overall.
A fault-tolerant quantum computer will be supported by a classical decoding system interfacing with quantum hardware to perform quantum error correction. It is important that the decoder can keep pace with the quantum clock speed, within the limitations on communication that are imposed by the physical architecture. To this end, we propose a local ``predecoder,'' which makes greedy corrections to reduce the amount of syndrome data sent to a standard matching decoder. We study these classical overheads for the surface code under a phenomenological phase-flip noise model with imperfect measurements. We find substantial improvements in the run time of the global decoder and the communication bandwidth by using the predecoder. For instance, to achieve a logical-failure probability of $f={10}^{\ensuremath{-}15}$ using qubits with physical error rate $p={10}^{\ensuremath{-}3}$ and a distance d=22d=22d=22 code, we find that the bandwidth cost is reduced by a factor of 1000 and that the time taken by a matching decoder is sped up by a factor of 200. To achieve this target failure probability, the predecoding approach requires a 50% increase in the qubit count compared with the optimal decoder.
Quantum error correction allows for quantum information to be preserved using logical qubits, which are subject to lower error rates than their constituent physical qubits. The degree of error suppression depends on the choice of error correcting code and distance, the underlying physical error rate, and the accuracy of the decoder. While traditional decoders utilise a binary (hard) syndrome, recent work shows that additional (soft) information captured during qubit readout can be effectively utilised to improve decoding accuracy. In this work, we present experimental results from a distance-three surface code implemented on transmon qubits, where we perform Z-stabiliser measurements to protect the state of the logical qubit against bit-flip errors. We initialise the logical qubit in one of 16 possible computational states representing the logical zero state, and perform repeated stabiliser checks over a variable number of rounds to preserve the state over time. We compare the decoding performance for a hard minimum-weight perfect matching decoder against a soft variant where rich measurement information is incorporated, and demonstrate an improved logical fidelity. Additionally, we employ a recurrent neural network decoder with both soft and hard variants and observe improved performance when soft information is used. The general nature of soft information makes it widely applicable to different physical qubit platforms, where it can be leveraged to shorten measurement times and improve the logical fidelity in quantum error correction experiments. Pre-print available at arXiv:2403.00706.
Quantum error correction (QEC) enables fault-tolerant quantum computation but requires operating quantum hardware at physical error rates below code-dependent thresholds, which remains challenging for current devices. We introduce syndrome resampling, a general method that increases QEC thresholds of any decoder and suppresses logical errors without additional hardware, decoding modifications, or code-specific assumptions beyond syndrome statistics. The method exploits the fact that syndromes with low probability are likely to lead to logical failure, therefore biasing syndrome averages towards most likely syndromes effectively increases logical fidelities. We establish a direct connection between the Rényi coherent information (RCI) and powers of the syndrome probability distribution, showing that resampling syndromes according to these powers combined with maximum likelihood decoding (MLD) realizes a family of optimal thresholds associated with phase transitions in the RCI. Numerical simulations of surface codes demonstrate that syndrome resampling substantially increases thresholds for both optimal and suboptimal decoders and reduces logical error rates by up to four orders of magnitude in experimentally relevant regimes. We further show that syndrome resampling can be effectively implemented from finite data and combined with decoding-based post-selection to achieve additional gains. Finally, applying the method to existing experimental QEC data yields up to two orders of magnitude reduction in logical error rates without requiring additional measurements. Our results provide a practical and decoder-agnostic route to improved logical fidelities in near-term QEC experiments.