Neural Minimum Weight Perfect Matching for Quantum Error Codes

TL;DR

This work tackles decoding in quantum error correction by augmenting the classical Minimum Weight Perfect Matching (MWPM) with a neural predictor that assigns dynamic edge weights based on syndrome data. The Neural Minimum Weight Perfect Matching (NMWPM) architecture combines a Graph Neural Network for local topology and a Transformer for global correlations, trained via a differentiable proxy loss to guide an MWPM step. Empirical results on Toric and Rotated Surface Codes under depolarizing and independent noise show meaningful reductions in logical error rate (LER) and competitive thresholds, with favorable parameter efficiency relative to strong baselines. The approach demonstrates the viability and practical impact of hybrid, data-driven decoders that preserve algorithmic structure while leveraging learned priors to exploit syndrome information. This work thus advances scalable, high-accuracy decoding for fault-tolerant quantum computation.

Abstract

Realizing the full potential of quantum computation requires Quantum Error Correction (QEC). QEC reduces error rates by encoding logical information across redundant physical qubits, enabling errors to be detected and corrected. A common decoder used for this task is Minimum Weight Perfect Matching (MWPM) a graph-based algorithm that relies on edge weights to identify the most likely error chains. In this work, we propose a data-driven decoder named Neural Minimum Weight Perfect Matching (NMWPM). Our decoder utilizes a hybrid architecture that integrates Graph Neural Networks (GNNs) to extract local syndrome features and Transformers to capture long-range global dependencies, which are then used to predict dynamic edge weights for the MWPM decoder. To facilitate training through the non-differentiable MWPM algorithm, we formulate a novel proxy loss function that enables end-to-end optimization. Our findings demonstrate significant performance reduction in the Logical Error Rate (LER) over standard baselines, highlighting the advantage of hybrid decoders that combine the predictive capabilities of neural networks with the algorithmic structure of classical matching.

Figures (7)

• Figure 1: Overview of the proposed decoding pipeline. (a) Three physical errors on the lattice generate four discrete syndrome defects. (b) These defects form the vertices of a complete graph used for matching. (c) The complete NMWPM architecture processes this graph structure to predict dynamic edge weights for the final correction.
• Figure 2: Comparison of LER vs. Physical Error Rate on the Toric Code under Depolarizing Noise
• Figure 3: Comparison of LER vs. Physical Error Rate on the Rotated Surface Code under Depolarizing Noise
• Figure 4: Error Threshold Analysis
• Figure 5: Evolution of predicted edge weight distributions for the Rotated Surface Code for $L_{code} = 7$. The transition to a bimodal distribution highlights the model's increasing confidence.