# Maximum Likelihood Decoding of Quantum Error Correction Codes

arXiv: https://arxiv.org/abs/2605.17230 · HTML: https://mlsanity.com/paper/2605.17230/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 99 references that are in the map.
- First posted on arXiv 17 May 2026.

## Builds on (most cited first, 11 of 11)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Learning to Decode the Surface Code with a Recurrent, Transformer-Based Neural Network](https://mlsanity.com/paper/2310.05900/) (2023)
- [Reinforcement Learning Decoders for Fault-Tolerant Quantum Computation](https://mlsanity.com/paper/1810.07207/) (2018)
- [Deep neural decoders for near term fault-tolerant experiments](https://mlsanity.com/paper/1802.06441/) (2018)
- [qecGPT: decoding Quantum Error-correcting Codes with Generative Pre-trained Transformers](https://mlsanity.com/paper/2307.09025/) (2023)
- [A scalable and real-time neural decoder for topological quantum codes](https://mlsanity.com/paper/2512.07737/) (2025)
- [Neural Network Decoders for Large-Distance 2D Toric Codes](https://mlsanity.com/paper/1809.06640/) (2018)
- [Efficient and Universal Neural-Network Decoder for Stabilizer-Based Quantum Error Correction](https://mlsanity.com/paper/2502.19971/) (2025)
- [Fusion Blossom: Fast MWPM Decoders for QEC](https://mlsanity.com/paper/2305.08307/) (2023)
- [Rethink the Role of Neural Decoders in Quantum Error Correction](https://mlsanity.com/paper/2605.12046/) (2026)
- [Learning to Decode in Parallel: Self-Coordinating Neural Network for Real-Time Quantum Error Correction](https://mlsanity.com/paper/2601.09921/) (2026)

## Built on by (most cited first, 1 of 1)

- [Efficient foundation decoders for fault-tolerant quantum computing](https://mlsanity.com/paper/2606.27119/) (2026)

## Abstract (arXiv)

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.