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ACM/IEEE Design Automation Conference (DAC)

Weighted Range-Constrained Ising-Model Decoder for Quantum Error Correction

Xinyi Guo, Hiromitsu Awano, Takashi G. Sato

Published Jun 22, 2025• 0 citations
Publisher

Abstract

Ising model-based Quantum Error Correction decoders reduce topological complexity compared to classical decoders. However, the SOTA Ising decoder has a higher time complexity than union-find (UF) and a lower threshold than minimum-weight perfect-matching (MWPM). We propose the Weighted Range-Constrained Ising Model-Based (WRIM) decoder. WRIM uses a polygonal region to enclose flipped syndromes, ensuring the coverage of all potential error chains while optimizing coupling and external field coefficients. WRIM reduces the variable count by $97.8 x$, achieves microsecondlevel decoding, and has a worst-case time complexity of $O(n)$, outperforming UF. WRIM exhibits threshold behavior up to 10.7$\mathbf{1 1. 0 \%}$, surpassing the MWPM’s highest reported threshold.

DOI
10.1109/dac63849.2025.11133309

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arXiv:2507.23618•Oct 26, 2025•0 citations
Ising model-based Quantum Error Correction decoders reduce topological complexity compared to classical decoders. However, the SOTA Ising decoder has a higher time complexity than union-find (UF) and a lower threshold than minimum-weight perfect-matching (MWPM). We propose the Weighted Range-Constrained Ising Model-Based (WRIM) decoder. WRIM uses a polygonal region to enclose flipped syndromes, ensuring the coverage of all potential error chains while optimizing coupling and external field coefficients. WRIM reduces the variable count by 97.8x97.8 x97.8x, achieves microsecondlevel decoding, and has a worst-case time complexity of O(n)O(n)O(n), outperforming UF. WRIM exhibits threshold behavior up to 10.711.0%\mathbf{1 1. 0 \%}11.0%, surpassing the MWPM’s highest reported threshold.
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.
Quantum error correction is needed for future quantum computers. Classical error correcting codes are not suitable for this due to the nature of quantum mechanics. Therefore, new codes need to be developed. A promising candidate is the toric code, a surface code, because of its locality and its high error correcting capability and thresholds (the error probability below which increasing the size of the code decreases the failure rate). This thesis provides an introduction to quantum error correction and the stabilizer formalism, which is used to introduce the toric code. Several decoders are looked at, including the Minimum Weight Perfect Matching (MWPM) decoder and a recently developed decoder, the Union-Find (UF) decoder. The UF decoder is useful because of its almost-linear time complexity, while only reducing the threshold by a marginal amount compared to the MWPM decoder. In this thesis, the time complexity of the weighted growth version of the UF decoder is analysed. The toric code and the decoders have been implemented and simulated, and the thresholds have been determined. For the MWPM decoder, the threshold was determined to be 11.5±0.2%, which is not in agreement with the threshold in literature of 10.3%, and should not be possible according to the optimal threshold of about 11%. This probably is a result of the low grid sizes (up to a gridsize of 11) of the code used due to the time needed to simulate the MWPM decoder. For the UF decoder, the threshold found was 9.7 ± 0.9%, which is in agreement with the threshold of 9.9% found by N. Delfosse and N. Nickerson. The time complexity of the UF decoder has also been determined, and is indeed almost-linear as expected by the analysis.
Realizing the full potential of quantum computation requires Quantum Error Correction (QEC). QEC reduces error rates by encoding logical information across redundant physical qubits, enabling errors to be detected and corrected. A common decoder used for this task is Minimum Weight Perfect Matching (MWPM) a graph-based algorithm that relies on edge weights to identify the most likely error chains. In this work, we propose a data-driven decoder named Neural Minimum Weight Perfect Matching (NMWPM). Our decoder utilizes a hybrid architecture that integrates Graph Neural Networks (GNNs) to extract local syndrome features and Transformers to capture long-range global dependencies, which are then used to predict dynamic edge weights for the MWPM decoder. To facilitate training through the non-differentiable MWPM algorithm, we formulate a novel proxy loss function that enables end-to-end optimization. Our findings on the toric code under depolarizing noise demonstrate thresholds of 17.9% and 10.95%, nearing the 18.9% and 11.0% maximum likelihood bounds, highlighting the advantage of hybrid decoders that combine the predictive capabilities of neural networks with the algorithmic structure of classical matching.
The union-find decoder is a leading algorithmic approach to the correction of quantum errors on the surface code, achieving code thresholds comparable to minimum-weight perfect matching (MWPM) with amortized computational time scaling near-linearly in the number of physical qubits. This complexity is achieved via optimizations provided by the disjoint-set data structure. We demonstrate, however, that the behavior of the decoder at scale underutilizes this data structure for twofold analytic and algorithmic reasons, and that improvements and simplifications can be made to architectural designs to reduce resource overhead in practice. To reinforce this, we model the behavior of erasure clusters formed by the decoder and show that there does not exist a percolation threshold within the data structure for any mode of operation. This yields a linear-time worst-case complexity for the decoder at scale, even with a naive implementation omitting popular optimizations. Published by the American Physical Society 2024
Quantum error correction is one of the most important milestones for the realization of large-scale quantum computation. To achieve this, it is essential not only to integrate a large number of qubits with high fidelity but also to build a scalable classical system that can perform error correction. Here, we propose an efficient and scalable decoder for quantum error correction using ``Fujitsu Digital Annealer'' (DA). Specifically, the error correction problem of stabilizer codes is mapped into an Ising-type optimization problem, so-called quadratic unconstrained binary optimization (QUBO) problem, which is solved by DA. In particular, we implement the proposed DA decoder for the surface code and perform detailed numerical experiments for various code distances to see its performance and scalability. We observe that computational scaling for the DA decoder has a lower order of polynomials than the decoding methods using simulated annealing (SA) and minimum-weight perfect matching (MWPM) algorithm under all tested conditions. It is also shown that the DA decoder has advantages over the Union-Find (UF) decoder from a variety of perspectives including hardware implementation. Furthermore, the threshold behavior of the logical error probability for the DA decoder is analyzed and the resultant threshold lies between 9.4% and 9.8%, which is very close to that obtained by the MWPM decoder. This result clearly shows the high potential of the DA decoder for quantum error correction.
Conventional quantum error correction (QEC) de-coders such as Minimum-Weight Perfect Matching (MWPM) and Union-Find (UF) offer high thresholds and fast decoding, respectively, but both suffer from high topological complexity. In contrast, Ising model-based decoders reduce topological complexity but demand considerable decoding time. We propose the Symmetric One-Hot Matching Elector (SOME), a novel decoder that reformulates the QEC decoding task as a Quadratic Unconstrained Binary Optimization (QUBO) problem—termed the One-Hot QUBO (OHQ). Each variable in the QUBO represents whether a given pair of flipped syndromes is matched, while the error probabilities between the pair are encoded as interaction coefficients (weight). Constraints ensure that each flipped syndrome is matched exactly once. Valid solutions of OHQ correspond to self-inverse permutation matrices, characterized by symmetric one-hot encoding. To solve the OHQ efficiently, SOME reformulates the decoding task as the construction of permutation matrices that minimize the total weight. It initializes each candidate matrix from one of the minimum-weight syndrome pairs, then iteratively appends additional pairs in ascending order of weight, and finally selects the permutation matrix with the lowest total energy. SOME achieves up to a 99.9x reduction in variable count and reduces decoding times from milliseconds to microseconds on a single-threaded commodity CPU. OHQ also maintains performance up to a 10.5% physical error rate, surpassing the highest known threshold of MWPM.