Maximum Likelihood Decoding of Quantum Error Correction Codes

2026-05-16

AI Summary

Although maximum-likelihood decoding (MLD) achieves optimal performance in quantum error correction, its practical application is hindered by #P-hard computational complexity. This work presents the first unified theoretical and computational framework for MLD by systematically integrating methods from statistical mechanics, tensor networks, and artificial intelligence. By leveraging the partition function of disordered spin models, tensor network contractions on factor graphs, and autoregressive generative models—including recurrent Transformers—the study reveals intrinsic connections among diverse decoding strategies. The proposed approximate decoders attain near-MLD accuracy while operating in polynomial time, demonstrating high performance and efficient parallel acceleration on both simulated and real quantum hardware.

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.

Problem

• quantum error correction
• maximum likelihood decoding
• computational intractability
• #P-hard
• fault-tolerant quantum computation

Innovation

• maximum likelihood decoding
• tensor networks
• statistical mechanics
• neural decoders
• quantum error correction