# Neural Network Decoders for Large-Distance 2D Toric Codes

arXiv: https://arxiv.org/abs/1809.06640 · HTML: https://mlsanity.com/paper/1809.06640/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Sep 2018.

## Built on by (most cited first, 8 of 8)

- [Learning to Decode the Surface Code with a Recurrent, Transformer-Based Neural Network](https://mlsanity.com/paper/2310.05900/) (2023)
- [A scalable and fast artificial neural network syndrome decoder for surface codes](https://mlsanity.com/paper/2110.05854/) (2021)
- [The END: An Equivariant Neural Decoder for Quantum Error Correction](https://mlsanity.com/paper/2304.07362/) (2023)
- [Scalable Neural Decoders for Practical Real-Time Quantum Error Correction](https://mlsanity.com/paper/2510.22724/) (2025)
- [Learning to Decode in Parallel: Self-Coordinating Neural Network for Real-Time Quantum Error Correction](https://mlsanity.com/paper/2601.09921/) (2026)
- [Maximum Likelihood Decoding of Quantum Error Correction Codes](https://mlsanity.com/paper/2605.17230/) (2026)
- [Efficient foundation decoders for fault-tolerant quantum computing](https://mlsanity.com/paper/2606.27119/) (2026)
- [Machine Learning Approaches to Decoding Topological Quantum Codes](https://mlsanity.com/paper/2608.15760/) (2026)

## Abstract (arXiv)

We still do not have perfect decoders for topological codes that can satisfy all needs of different experimental setups. Recently, a few neural network based decoders have been studied, with the motivation that they can adapt to a wide range of noise models, and can easily run on dedicated chips without a full-fledged computer. The later feature might lead to fast speed and the ability to operate at low temperatures. However, a question which has not been addressed in previous works is whether neural network decoders can handle 2D topological codes with large distances. In this work, we provide a positive answer for the toric code. The structure of our neural network decoder is inspired by the renormalization group decoder. With a fairly strict policy on training time, when the bit-flip error rate is lower than 9% and syndrome extraction is perfect, the neural network decoder performs better when code distance increases. With a less strict policy, we find it is not hard for the neural decoder to achieve a performance close to the minimum-weight perfect matching algorithm. The numerical simulation is done up to code distance d=64. Last but not least, we describe and analyze a few failed approaches. They guide us to the final design of our neural decoder, but also serve as a caution when we gauge the versatility of stock deep neural networks. The source code of our neural decoder can be found at https://github.com/XiaotongNi/toric-code-neural-decoder .