Generalized matching decoders for 2D topological translationally-invariant codes

2026-03-05

AI Summary

This work addresses the lack of efficient, high-performance decoders for two-dimensional translation-invariant topological quantum codes, such as bivariate bicycle codes. The authors propose a novel decoding framework based on coarse-grained mapping and graph matching, which maps the syndrome of a general translation-invariant code to an equivalent surface code excitation pattern. This approach extends graph-matching decoding—previously limited to specific models—to this broad class of topological codes for the first time. Theoretical analysis establishes that the decoder possesses provable error-correction capabilities and a non-zero code-capacity threshold. Numerical experiments demonstrate its ability to correct errors with weight up to a constant fraction of the code distance, achieving performance on bivariate bicycle codes comparable to belief propagation combined with ordered statistics decoding.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Two-dimensional topological translationally-invariant (TTI) quantum codes, such as the toric code (TC) and bivariate bicycle (BB) codes, are promising candidates for fault-tolerant quantum computation. For such codes to be practically relevant, their decoders must successfully correct the most likely errors while remaining computationally efficient. For the TC, graph-matching decoders satisfy both requirements and, additionally, admit provable performance guarantees. Given the equivalence between TTI codes and (multiple copies of) the TC, one may then ask whether TTI codes also admit analogous graph-matching decoders. In this work, we develop a graph-matching approach to decoding general TTI codes. Intuitively, our approach coarse-grains the TTI code to obtain an effective description of the syndrome in terms of TC excitations, which can then be removed using graph-matching techniques. We prove that our decoders correct errors of weight up to a constant fraction of the code distance and achieve non-zero code-capacity thresholds. We further numerically study a variant optimized for practically relevant BB codes and observe performance comparable to that of the belief propagation with ordered statistics decoder. Our results indicate that graph-matching decoders are a viable approach to decoding BB codes and other TTI codes.

Problem

• topological codes
• translationally-invariant codes
• quantum error correction
• decoding
• graph matching

Innovation

• graph-matching decoder
• topological quantum codes
• translational invariance
• coarse-graining
• fault-tolerant quantum computation