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arXiv:2510.06531 · Physical Review A

Approximate maximum-likelihood decoding with K minimum weight matchings

Mao Lin

Published Oct 6, 2025• 0 citations
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Abstract

Minimum weight matching (MWM) and maximum likelihood decoding (MLD) are two widely used and distinct decoding strategies for quantum error correction. For a given syndrome, the MWM decoder finds the most probable physical error corresponding to the MWM of the decoding graph, whereas MLD aims to find the most probable logical error. Although MLD is the optimal error correction strategy, it is typically more computationally expensive compared to the MWM decoder. In this work, we introduce an algorithm that approximates MLD with $K$ MWMs from the decoding graph. Taking the surface code subject to graphlike errors as an example, we show that it is possible to efficiently find the first $K$ MWMs by systematically modifying the original decoding graph followed by finding the MWMs of the modified graphs. For the case where the $X$ and $Z$ errors are correlated, despite the fact that the MWM of the decoding hypergraph cannot be found efficiently, we present a heuristic approach to approximate the MLD by finding the $K$ MWMs in the $X$ and $Z$ subgraphs. We benchmark the efficacy of our algorithm for the surface code subject to graphlike errors as well as the surface-square Gottesman-Kitaev-Preskill (GKP) code and surface-hexagonal GKP code subject to the Gaussian random displacement errors, showing that the fidelity approaches that of the exact MLD (for the first two cases) or the tensor-network decoder (for the last case) as $K$ increases.

DOI
10.1103/vf9r-z5qg

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Minimum weight matching (MWM) and maximum likelihood decoding (MLD) are two widely used and distinct decoding strategies for quantum error correction. For a given syndrome, the MWM decoder finds the most probable physical error corresponding to the MWM of the decoding graph, whereas MLD aims to find the most probable logical error. Although MLD is the optimal error correction strategy, it is typically more computationally expensive compared to the MWM decoder. In this work, we introduce an algorithm that approximates MLD with KKK MWMs from the decoding graph. Taking the surface code subject to graphlike errors as an example, we show that it is possible to efficiently find the first KKK MWMs by systematically modifying the original decoding graph followed by finding the MWMs of the modified graphs. For the case where the XXX and ZZZ errors are correlated, despite the fact that the MWM of the decoding hypergraph cannot be found efficiently, we present a heuristic approach to approximate the MLD by finding the KKK MWMs in the XXX and ZZZ subgraphs. We benchmark the efficacy of our algorithm for the surface code subject to graphlike errors as well as the surface-square Gottesman-Kitaev-Preskill (GKP) code and surface-hexagonal GKP code subject to the Gaussian random displacement errors, showing that the fidelity approaches that of the exact MLD (for the first two cases) or the tensor-network decoder (for the last case) as KKK increases.
Maximum-likelihood decoding (MLD) achieves the minimum logical error rate of stabilizer codes under known i.i.d. Pauli noise, but its exact evaluation is \#P-hard. Practical pipelines therefore approximate MLD by minimum-weight decoding (MWD), retaining only the lowest-weight recovery per syndrome and discarding the coset degeneracy. The minimum-weight search is in turn implemented by stochastic solvers. We introduce approximate maximum-likelihood decoding (AMLD), a black-box framework that recycles the candidate samples discarded by stochastic inner decoders into a per-class truncated free-energy estimator. For every logical class represented in the candidate pool, the estimator is provably bounded below by the exact free energy and above by the empirical minimum weight. AMLD returns the logical class minimizing the estimated free energy with linear classical overhead. In SA-based Ising-decoder benchmarks, AMLD closes up to 83%83\%83% of the MWD--MLD threshold gap across the toric and color codes under bit-flip and depolarizing noise. The largest threshold improvement, from 17.28%17.28\%17.28% to 18.62%18.62\%18.62%, occurs on the 6.6.66.6.66.6.6 color code under depolarizing noise. We further demonstrate AMLD on the [[144,12,12]][[144,12,12]][[144,12,12]] bivariate-bicycle code, whose bit-flip decoding problem has a hypergraph structure. This application requires neither matching-based enumeration nor code-specific tensor-network contraction. At p=0.05p=0.05p=0.05, AMLD reduces the logical error rate by 13%13\%13% relative to MWD evaluated on the same BP-OSD candidate pool.
We describe two implementations of the optimal error correction algorithm known as the maximum likelihood decoder (MLD) for the 2D surface code with a noiseless syndrome extraction. First, we show how to implement MLD exactly in time O(n2)O(n^2)O(n2), where nnn is the number of code qubits. Our implementation uses a reduction from MLD to simulation of matchgate quantum circuits. This reduction however requires a special noise model with independent bit-flip and phase-flip errors. Secondly, we show how to implement MLD approximately for more general noise models using matrix product states (MPS). Our implementation has running time O(nχ3)O(nχ^3)O(nχ3) where χχχ is a parameter that controls the approximation precision. The key step of our algorithm, borrowed from the DMRG method, is a subroutine for contracting a tensor network on the two-dimensional grid. The subroutine uses MPS with a bond dimension χχχ to approximate the sequence of tensors arising in the course of contraction. We benchmark the MPS-based decoder against the standard minimum weight matching decoder observing a significant reduction of the logical error probability for χ≥4χ\ge 4χ≥4.
Quantum error correction (QEC) is indispensable for realizing fault-tolerant quantum computation, yet its effectiveness hinges critically on the classical decoding algorithm that interprets noisy syndrome measurements. Among all possible decoding strategies, maximum likelihood decoding (MLD) is provably optimal, since it identifies the logical group with largest likelihood by summing over all possible errors within logical class consistent with the observed syndrome. Despite its optimality, MLD is computationally intractable in general (#P-hard), motivating a rich landscape of exact and approximate algorithms. In this topical review, we provide a unified perspective on MLD by surveying recent advances through three complementary lenses: statistical mechanics, tensor networks, and artificial intelligence. From the statistical mechanics viewpoint, the MLD problem maps onto evaluating partition functions of disordered spin models, enabling exact solutions for certain codes and noise models as well as threshold estimation via phase-transition analysis. From the tensor network perspective, approximate contraction of tensor networks on the code's factor graph yields decoders that closely approach MLD accuracy with polynomial computational cost. From the artificial intelligence perspective, neural-network-based decoders, including autoregressive generative models and recurrent transformers, learn to approximate the MLD distribution from data, achieving high accuracy with the parallelism afforded by modern hardware accelerators. We discuss the connections among these three approaches, review their application to both simulated and experimental quantum hardware, and outline open challenges including real-time decoding, scalability to large code distances, and generalization to high-rate quantum low-density parity-check codes.
Maximum likelihood decoding (MLD) achieves optimal accuracy but is computationally infeasible for large-distance codes. Tensor-network-based MLD (TN-MLD) improves efficiency but has so far only been shown to work for small-distance surface codes. We introduce a hypergraph-based approximate MLD (HAMLD) that enables more efficient decoding of more general codes under circuit-level noise. Unlike TN-MLD, our HAMLD employs syndrome-guided hyperedge selection and probability-based truncation to reduce redundant computation. We benchmarked our approach on surface codes, color codes, and bivariate-bicycle (BB) codes under a circuit-level noise model. Numerical results demonstrate that HAMLD can decode BB codes with hundreds of qubits, which MLD and TN-MLD cannot handle. For a surface code with distance and repetition rounds both equal to 3, HAMLD achieves optimal decoding accuracy while being about 323 × faster than MLD and about 551 × faster than TN-MLD. HAMLD also achieves higher accuracy than common decoders—minimum-weight perfect matching, Chromobius, and BP + OSD —in the two smallest parameter settings we considered, reducing logical error rates by an average of 27.8% for surface codes, 58.5% for color codes, and 74.4% for BB codes through degeneracy-aware and precise hypergraph selection.
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.
Quantum error correction (QEC) plays an essential role in fault-tolerantly realizing quantum algorithms of practical interest.Among different approaches to QEC, encoding logical quantum information in harmonic oscillator modes has been shown to be promising and hardware efficient.In this work, we study multimode Gottesman-Kitaev-Preskill (GKP) codes, encoding a qubit in many oscillators, through a lattice perspective.In particular, we implement a closest point decoding strategy for correcting random Gaussian shift errors.For decoding a generic multimode GKP code, we first identify its corresponding lattice followed by finding the closest lattice point in its symplectic dual lattice to a candidate shift error compatible with the error syndrome.We use this method to characterize the error-correction capabilities of several known multimode GKP codes, including their code distances and fidelities.We also perform numerical optimization of multimode GKP codes up to ten modes and find three instances (with three, seven, and nine modes) with better code distances and fidelities compared to the known GKP codes with the same number of modes.While exact closest point decoding incurs exponential time cost in the number of modes for general unstructured GKP codes, we give several examples of structured GKP codes (i.e., of the repetition-rectangular GKP code types) where the closest point decoding can be performed exactly in linear time.For the surface-GKP code, we show that the closest point decoding can be performed exactly in polynomial time with the help of a minimum-weight-perfect-matching algorithm (MWPM).We show that this MWPM closest point decoder improves both the fidelity and the noise threshold of the surface-GKP code to 0.602 compared to the previously studied MWPM decoder assisted by log-likelihood analog information, which yields a noise threshold of 0.599.