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arXiv:2212.11632 · Physical Review A

Performance enhancement of surface codes via recursive minimum-weight perfect-match decoding

Antonio deMarti iOlius, Josu Etxezarreta Martínez, Patricio Fuentes, Pedro M. Crespo

Published Aug 3, 2023• 17 citations
arXivHTMLPDF

Abstract

The minimum weight perfect matching (MWPM) decoder is the standard decoding strategy for quantum surface codes. However, it suffers a harsh decrease in performance when subjected to biased or nonidentical quantum noise. In this work, we modify the conventional MWPM decoder so that it considers the biases, the nonuniformities, and the relationship between $X$, $Y$, and $Z$ errors of the constituent qubits of a given surface code. Our modified approach, which we refer to as the recursive MWPM decoder, obtains an $18%$ improvement in the probability threshold ${p}_{\text{th}}$ under depolarizing noise. We also obtain significant performance improvements when considering biased noise and independent nonidentically distributed (i.ni.d.) error models derived from measurements performed on state-of-the-art quantum processors. In fact, when subjected to i.ni.d. noise, the recursive MWPM decoder yields a performance improvement of $105.5%$ over the conventional MWPM strategy, and in some cases, it even surpasses the performance obtained over the well-known depolarizing channel.

DOI
10.1103/physreva.108.022401

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The minimum weight perfect matching (MWPM) decoder is the standard decoding strategy for quantum surface codes. However, it suffers a harsh decrease in performance when subjected to biased or nonidentical quantum noise. In this work, we modify the conventional MWPM decoder so that it considers the biases, the nonuniformities, and the relationship between XXX, YYY, and ZZZ errors of the constituent qubits of a given surface code. Our modified approach, which we refer to as the recursive MWPM decoder, obtains an 1818%18 improvement in the probability threshold pth{p}_{\text{th}}pth​ under depolarizing noise. We also obtain significant performance improvements when considering biased noise and independent nonidentically distributed (i.ni.d.) error models derived from measurements performed on state-of-the-art quantum processors. In fact, when subjected to i.ni.d. noise, the recursive MWPM decoder yields a performance improvement of 105.5105.5%105.5 over the conventional MWPM strategy, and in some cases, it even surpasses the performance obtained over the well-known depolarizing channel.
Quantum Surface codes are a kind of quantum topological stabilizer codes whose stabilizers and qubits are geometrically related. Due to their special structures, surface codes have great potential to lead people to large-scale quantum computation. In the minimum weight perfect matching (MWPM) decoding of surface codes, the bit-flip errors and phase-flip errors are assumed to be independent for simplicity. However, these two kinds of errors are likely to be correlated in the real world. In this paper, we propose a modification to MWPM decoding for surface codes to deal with the noises in depolarizing channels where bit-flip errors and phase-flip errors are correlated. With this modification, we obtain thresholds of 17% and 15.3% for the surface codes with mixed boundaries and the surface codes with a hole, respectively
The minimum-weight perfect matching (MWPM) decoder is a standard decoding strategy for surface codes, but its performance can degrade considerably under biased noise. In this work, a modified family of surface codes, termed the XYZ planar code, is introduced, and MWPM is extended to posterior MWPM (pMWPM), which exploits the information provided by mixed stabilizers with almost no increase in decoding complexity. The XYZ--pMWPM and XZZX code--decoder combinations are compared under homogeneous independent and identically distributed (IID) Pauli noise and processor-derived independent nonidentically distributed (INID) noise. Under homogeneous biased Pauli noise, pMWPM substantially increases the threshold of the XYZ planar code relative to conventional MWPM. At finite code distances, particularly for small codes under strong bias, XYZ--pMWPM achieves logical error rates comparable to or lower than those of XZZX. Under INID noise, the pseudothreshold ordering observed under the corresponding homogeneous IID references is not always preserved; in some hardware--distance settings, XYZ--pMWPM achieves pseudothresholds comparable to or higher than those of XZZX. Within XZZX, qubit-specific weighting consistently yields higher INID pseudothresholds than average-rate weighting for the hardware instances considered. It is further found that the IID--INID pseudothreshold shift depends strongly on the physical-error scalarization and the construction of the homogeneous IID reference. These results show that pMWPM efficiently exploits the mixed-stabilizer structure of the XYZ planar code and provides competitive finite-distance decoding under biased and spatially nonuniform noise.
Efficient and accurate decoders are essential for quantum error correction. Surface codes are due to their high error threshold and scalability. The Minimum Weight Perfect Matching (MWPM) decoder and Union-Find (UF) decoder are the two most commonly used decoders for quantum surface codes, known for their accuracy and decoding speed, respectively. In this paper, we design an iteratively reweighted UF (IRUF) decoder considering the correlation of bit flip error and phase flip error in the depolarization noise channel. It has higher decoding accuracy than the MWPM decoder and UF decoder, and the decoding threshold is 11.5% and 6.3% higher respectively. Notably the IRUF decoder almost retains the same linear time complexity as the UF decoder, which is far better than the MWPM decoder.
Surface codes exploit topological protection to increase error resilience in quantum computing devices and can in principle be implemented in existing hardware. They are one of the most promising candidates for active error correction, not least due to a polynomial-time decoding algorithm which admits one of the highest predicted error thresholds. We consider the dependence of this threshold on underlying assumptions including different noise models, and analyze the performance of a minimum weight perfect matching (MWPM) decoding compared to a mathematically optimal maximum likelihood (ML) decoding. Our ML algorithm tracks the success probabilities for all possible corrections over time and accounts for individual gate failure probabilities and error propagation due to the syndrome measurement circuit. We present the very first evidence for the true error threshold of an optimal circuit level decoder, allowing us to draw conclusions about what kind of improvements are possible over standard MWPM.
The standard approach to Belief Propagation (BP) decoding passes messages on the Tanner graph of the quantum error-correcting (QEC) code, and is known to lack a threshold for the surface code. We propose novel BP decoders that exchange messages on the decoding graph and obtain thresholds (both for code capacity and circuit-level noise) via standalone BP for the surface code under depolarizing noise. Our approach, similarly to the minimum weight perfect matching (MWPM) decoder, is applicable to any graphlike QEC code. The thresholds observed with our decoders are close to those obtained by MWPM. This result opens the path towards scalable hardware-accelerated implementations of MWPM-compatible decoders.
Quantum hardware suffers from high error rates and noise, which makes directly running applications on them ineffective. Quantum Error Correction (QEC) is a critical technique towards fault tolerance which encodes the quantum information distributively in multiple data qubits and uses syndrome qubits to check parity. Minimum-Weight-Perfect-Matching (MWPM) is a popular QEC decoder that takes the syndromes as input and finds the matchings between syndromes that infer the errors. However, there are two paramount challenges for MWPM decoders. First, as noise in real quantum systems can drift over time, there is a potential misalignment with the decoding graph's initial weights, leading to a severe performance degradation in the logical error rates. Second, while the MWPM decoder addresses independent errors, it falls short when encountering correlated errors typical on real hardware, such as those in the 2Q depolarizing channel. We propose DGR, an efficient decoding graph edge re-weighting strategy with no quantum overhead. It leverages the insight that the statistics of matchings across decoding iterations offer rich information about errors on real quantum hardware. By counting the occurrences of edges and edge pairs in decoded matchings, we can statistically estimate the up-to-date probabilities of each edge and the correlations between them. The reweighting process includes two vital steps: alignment re-weighting and correlation re-weighting. The former updates the MWPM weights based on statistics to align with actual noise, and the latter adjusts the weight considering edge correlations. Extensive evaluations on surface code and honeycomb code under various settings show that DGR reduces the logical error rate by 3.6x on average-case noise mismatch with exceeding 5000x improvement under worst-case mismatch.