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arXiv:2609.03928

Approximate maximum-likelihood decoding via truncated free energies

Yuanqi Liu, Weilei Zeng, Junjie Wu, Lingling Lao

quant-phSubmitted Sep 3, 2026• Announced Sep 4, 2026• 0 citations
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Abstract

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 $83\%$ of the MWD--MLD threshold gap across the toric and color codes under bit-flip and depolarizing noise. The largest threshold improvement, from $17.28\%$ to $18.62\%$, occurs on the $6.6.6$ color code under depolarizing noise. We further demonstrate AMLD on the $[[144,12,12]]$ 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 $p=0.05$, AMLD reduces the logical error rate by $13\%$ relative to MWD evaluated on the same BP-OSD candidate pool.

Comments
12 pages, 7 figures

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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.
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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.
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.
Quantum low-density parity-check (qLDPC) codes reduce the qubit overhead of fault-tolerant quantum computation by an order of magnitude, but their decoding is harder than its classical counterpart: because many physical errors are equivalent up to stabilizers, the degenerate maximum-likelihood (ML) decoder must compare the probabilities of entire equivalence classes of errors, that is, partition functions, rather than single errors. The workhorse decoder BP+OSD sidesteps degeneracy heuristically and offers no guarantees. We treat degenerate decoding as probabilistic inference in an undirected graphical model: the probability of each logical class is the partition function of an unconstrained, strictly positive Markov random field over the code's check variables, a construction that generalizes the random-bond Ising mapping of the surface code to arbitrary CSS codes and to spacetime decoding with measurement errors and circuit-level noise. On this model we build two decoders. The first estimates all class partition functions by annealed importance sampling with common random numbers and attaches to every decision a certificate of optimality: a paired bootstrap test, or, composed with constant-factor estimators such as WISH, an exact optimality proof. The second is region-based: the Bethe free energy, whose bias cancels between classes, reproduces exact ML decoding on every tested surface-code instance at millisecond cost, and enlarging the regions to elimination clusters makes exact degenerate ML decoding of the [[72,12,6]] bivariate bicycle code feasible. Across surface codes and the bivariate bicycle codes [[72,12,6]] and [[144,12,12]], under code-capacity, phenomenological, and circuit-level noise, the sampling decoder matches or exceeds BP+OSD while certifying the bulk of its decisions, and the certificate flags exactly the syndromes on which any fast decoder should be distrusted.
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Maximum-likelihood decoding provides an optimal decoding strategy for quantum error correction under stochastic Pauli noise. However, computing logical-class probabilities is challenging, and the leading exact tensor-network contraction requires time exponential in treewidth. In this work, we introduce a new decoding algorithm based on rank-decomposition dynamic programming (Rank DP). We express decoding partition functions with independent local fault factors as quadratic sums of powers and apply Rank DP. The resulting exact evaluator has arithmetic complexity polynomial in the input size and exponential in the Tanner graph's rank-width with respect to the fault partition, including decomposition construction. After Gaussian elimination, it gives polynomial-time decoding for punctured quantum Reed-Muller codes and a Steane-concatenated family with growing distance under independent single-qubit Pauli noise. Standard contraction of the corresponding tensor networks requires superpolynomial time. A nonnegative realization gives relative floating-point error bounds under explicit arithmetic assumptions. Numerical experiments demonstrate runtime advantages over the tested tensor-network implementations on selected code-capacity and circuit-level instances, with full likelihood evaluation for Reed-Muller codes up to 1,0231{,}0231,023 qubits. We further use these likelihoods to learn circuit noise parameters from syndromes, evaluate rare postselection probabilities, and quantify decoder optimality gaps. Our work opens new avenues for exploiting algebraic structure in quantum decoding and noise characterization.