# The END: An Equivariant Neural Decoder for Quantum Error Correction

arXiv: https://arxiv.org/abs/2304.07362 · HTML: https://mlsanity.com/paper/2304.07362/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 28 references that are in the map.
- First posted on arXiv 14 Apr 2023.

## Builds on (most cited first, 7 of 7)

- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Super-Convergence: Very Fast Training of Neural Networks Using Large Learning Rates](https://mlsanity.com/paper/1708.07120/) (2017)
- [Neural Enhanced Belief Propagation on Factor Graphs](https://mlsanity.com/paper/2003.01998/) (2020)
- [Neural Network Decoders for Large-Distance 2D Toric Codes](https://mlsanity.com/paper/1809.06640/) (2018)

## Built on by (most cited first, 6 of 6)

- [Learning to Decode the Surface Code with a Recurrent, Transformer-Based Neural Network](https://mlsanity.com/paper/2310.05900/) (2023)
- [Scalable Neural Decoders for Practical Real-Time Quantum Error Correction](https://mlsanity.com/paper/2510.22724/) (2025)
- [Enhancing Quantum Memory Lifetime with Measurement-Free Local Error Correction and Reinforcement Learning](https://mlsanity.com/paper/2408.09524/) (2024)
- [Machine Learning Approaches to Decoding Topological Quantum Codes](https://mlsanity.com/paper/2608.15760/) (2026)
- [Equivariant Machine Learning Decoder for 3D Toric Codes](https://mlsanity.com/paper/2409.04300/) (2024)
- [When Quantum Meets AI: Quantum Methods for Machine Learning and Machine Learning Methods for Quantum Systems](https://mlsanity.com/paper/2609.25641/) (2026)

## Abstract (arXiv)

Quantum error correction is a critical component for scaling up quantum computing. Given a quantum code, an optimal decoder maps the measured code violations to the most likely error that occurred, but its cost scales exponentially with the system size. Neural network decoders are an appealing solution since they can learn from data an efficient approximation to such a mapping and can automatically adapt to the noise distribution. In this work, we introduce a data efficient neural decoder that exploits the symmetries of the problem. We characterize the symmetries of the optimal decoder for the toric code and propose a novel equivariant architecture that achieves state of the art accuracy compared to previous neural decoders.