Neural Belief-Matching Decoding for Topological Quantum Error Correction Codes

TL;DR

This paper introduces a neural belief-matching decoder for topological quantum error correction codes, reducing decoding complexity and training costs while maintaining high performance across different code sizes.

Contribution

It proposes a neural belief-matching decoder with a convolutional architecture that enables scalable training and transferability for topological quantum codes.

Findings

Neural belief-matching decoder reduces decoding complexity.

Convolutional architecture allows transfer to larger codes.

Maintains decoding performance across code sizes.

Abstract

Quantum error correction (QEC) is critical for scalable fault-tolerant quantum computing. Topological codes, such as the toric code, offer hardware-efficient architectures but their Tanner graphs contain many girth-4 cycles that degrade the performance of belief-propagation (BP) decoding. For this reason, BP decoding is typically followed by a more complex second stage decoder such as minimum-weight perfect matching. These combined decoders achieve a remarkable performance, albeit at the cost of increased complexity. In this paper we propose two key improvements for the decoding of toric code. The first one is replacing the BP decoder by a neural BP decoder, giving rise to the neural belief-matching decoder which substantially decreases the average decoding complexity. The main drawback of this approach is the high cost associated with the training of the neural BP decoder. To address this issue, we impose a convolutional architecture on the neural BP decoder, enabling weight sharing across the spatially homogeneous structure of the code's factor graph. This design allows a model trained on a modest-size topological code to be directly transferred to much larger instances, preserving decoding quality while dramatically lowering the training burden. Our numerical experiments on toric-code lattices of various sizes demonstrate that this technique does not result in a noticeable loss in performance.