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arXiv:2605.17230

Maximum Likelihood Decoding of Quantum Error Correction Codes

Hanyan Cao, Ge Yan, Yuxuan Du, Feng Pan

quant-phcond-mat.dis-nncond-mat.stat-mechcs.LGSubmitted May 17, 2026• Announced May 19, 2026• 0 citations
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Abstract

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.

Comments
An invited topical review. Comments are welcome

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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.
As quantum computers remain susceptible to noise, QEC is essential for preserving logical information during computations. However, the performance of QEC codes breaks down beyond certain noise thresholds, revealing fundamental limits on their ability to protect quantum information. These limits can be characterized using information-theoretic measures such as the coherent information, which quantifies the maximum rate at which logical information can be reliably transmitted through a noisy quantum channel. In this work, we establish a direct connection between the CI and the binary cross-entropy loss used when training neural network decoders. Specifically, we show that the CI constitutes a sharp lower bound on the achievable loss for decoders that track logical operators across noisy channels. To this end, we develop a transformer-based neural network model based on maximum likelihood decoding. We train this network to estimate the CI and evaluate its performance on the surface code under three noise models: code capacity, phenomenological, and circuit-level noise. Our results demonstrate that the network accurately predicts CI and yields threshold estimates that closely match known theoretical limits. When used as a decoder, the network significantly outperforms the minimum weight perfect matching decoder in terms of logical error rate. We also introduce a novel soft post-selection scheme that independently treats uncertainty in both logical operators and relies on confidence-based filtering of the network's output. We prove that such post-selection strategies, based on the MLD cosets, are optimal, and demonstrate their scalability in terms of both logical error rate and abort probability. These findings establish transformer-based architectures as powerful tools for QEC and provide the first numerical evidence supporting the optimality and scalability of MLD-based post-selection.
Efficient and accurate decoding of quantum error-correcting codes is essential for fault-tolerant quantum computation, however, it is challenging due to the degeneracy of errors, the complex code topology, and the large space for logical operators in high-rate codes. In this work, we propose a decoding algorithm utilizing generative modeling in machine learning. We employ autoregressive neural networks to learn the joint probability of logical operators and syndromes in an unsupervised manner, eliminating the need for labeled training data. The learned model can approximately perform maximum likelihood decoding by directly generating the most likely logical operators for kkk logical qubits with O(2k)\mathcal O(2k)O(2k) computational complexity. Thus, it is particularly efficient for decoding high-rate codes with many logical qubits. The proposed approach is general and applies to a wide spectrum of quantum error-correcting codes including surface codes and quantum low-density parity-check codes (qLDPC), under noise models ranging from code capacity noise to circuit level noise. We conducted extensive numerical experiments to demonstrate that our approach achieves significantly higher decoding accuracy compared to the minimum weight perfect matching and belief propagation with ordered statistics on the surface codes and high-rate quantum low-density parity-check codes. Our approach highlights generative artificial intelligence as a potential solution for the real-time decoding of realistic and high-rate quantum error correction codes.
Decoding is an essential component of quantum error correction (QEC), translating stabilizer measurement outcomes into corrective actions that suppress logical errors and preserve logical quantum information. Building fault-tolerant architectures requires increasing the code distance, which in turn places growing demands on decoding accuracy, scalability, and practical deployability. While a wide range of decoding algorithms have been proposed and demonstrated, achieving reliable, scalable, and real-time decoding remains a significant challenge. Machine-learning (ML) approaches are particularly well suited to this setting, as quantum error decoding is fundamentally a problem of processing large volumes of classical data with complex spatiotemporal correlations. This chapter surveys ML-based methods for quantum error decoding, with a focus on topological codes and an emphasis on architectural principles, practical performance, and real-time considerations. We first frame decoding as a learning problem and outline key paradigms, including discriminative, generative, and reinforcement-learning formulations. We then introduce the neural network building blocks that underpin most contemporary neural decoders and discuss how these components can be integrated to balance expressivity, scalability, and latency. Building on this architectural perspective, we review recent progress and benchmarks in neural decoding for memory experiments, and discuss real-time decoding, open challenges, and future directions toward scalable fault-tolerant quantum computing.
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.
We propose a general framework for decoding quantum error-correcting codes with generative modeling. The model utilizes autoregressive neural networks, specifically Transformers, to learn the joint probability of logical operators and syndromes. This training is in an unsupervised way, without the need for labeled training data, and is thus referred to as pre-training. After the pre-training, the model can efficiently compute the likelihood of logical operators for any given syndrome, using maximum likelihood decoding. It can directly generate the most-likely logical operators with computational complexity O(2k)\mathcal O(2k)O(2k) in the number of logical qubits kkk, which is significantly better than the conventional maximum likelihood decoding algorithms that require O(4k)\mathcal O(4^k)O(4k) computation. Based on the pre-trained model, we further propose refinement to achieve more accurately the likelihood of logical operators for a given syndrome by directly sampling the stabilizer operators. We perform numerical experiments on stabilizer codes with small code distances, using both depolarizing error models and error models with correlated noise. The results show that our approach provides significantly better decoding accuracy than the minimum weight perfect matching and belief-propagation-based algorithms. Our framework is general and can be applied to any error model and quantum codes with different topologies such as surface codes and quantum LDPC codes. Furthermore, it leverages the parallelization capabilities of GPUs, enabling simultaneous decoding of a large number of syndromes. Our approach sheds light on the efficient and accurate decoding of quantum error-correcting codes using generative artificial intelligence and modern computational power.
We study the exact maximum-likelihood decoding of quantum error-correcting codes. Our primary focus is on the repetition code under circuit-level noise, which serves as a fundamental basis for quantum error correction experiments and is the only code that has achieved large distances and extremely low error rates. We uncover that the repetition code under circuit-level noise has an exact solution and propose an optimal maximum-likelihood decoding algorithm called "planar." This algorithm is based on the exact solution of the spin-glass partition function on planar graphs and has polynomial computational complexity. Through extensive numerical experiments, we demonstrate that our algorithm uncovers the exact threshold for depolarizing noise and realistic superconductor SI1000 noise. Furthermore, we apply our method to analyze data from recent quantum memory experiments conducted by Google Quantum AI, revealing that part of the error floor was attributed to the decoding algorithm used by Google. Finally, we implemented the repetition code quantum memory on superconducting systems using a 72-qubit quantum chip lacking reset gates, demonstrating that even with an unknown error model, the proposed algorithm achieves a significantly lower logical error rate than the matching-based algorithm. We also discuss the application of our approach to quantum codes whose maximum-likelihood decoding problem corresponds to a spin-glass problem, including the surface code under independent code-capacity noise.