Towards Scalable Quaternary Message-Passing Decoding for Quantum Error Correction

2026-05-22

AI Summary

This work addresses the poor scalability and lack of theoretical grounding of existing message-passing decoders in quantum error correction at large code distances. The authors propose a more principled and interpretable quaternary message-passing decoding framework, integrating graph sparsification and complexity optimization strategies to significantly enhance decoding performance. The method achieves an apparent depolarizing threshold of 16% for code distances up to 20, outperforming minimum-weight perfect matching. Under pure X noise, it surpasses BP-OSD for the first time at code distance 65 and exhibits signs of an asymptotic threshold near 9%. Furthermore, graph-theoretic analysis provides theoretical justification for the algorithm’s scalability, effectively bridging the gap between practical performance and theoretical understanding.

Abstract

The scalability and interpretability of message-passing (MP) decoding, such as (quaternary) Belief Propagation, remain open challenges in quantum error correction. Even for surface codes, arguably the first testbed for decoding methods, studies of improved MP decoders have mostly been restricted to small distances ($d \lesssim 19$). Moreover, the mismatch with established message-passing theory limits the decoder's interpretability, making it unclear whether MP decoding can sustain its effectiveness at large system sizes. This work takes a step toward a more principled and interpretable MP decoding framework, with the goal of making MP-based decoding more reliable and bridging theory and practice. We introduce a dilution method, which allows a quaternary Min-Sum (MS) decoder to exhibit an apparent depolarizing threshold of $16\%$ up to distance $20$, outperforming Minimum-Weight Perfect Matching in finite-length regimes. Notably, for $X$-noise, the standard MS decoder under dilution has worst-case complexity $O(N \log^2 d)$ and outperforms BP-OSD at $d=65$. The observed $\sim 9\%$ threshold may correspond to a true asymptotic threshold. Finally, we give a graph-dilution argument that interprets the success of the dilution method and offers insight into when MP algorithms can genuinely scale. Taken together, these results provide encouraging progress toward scalable and interpretable MP decoding in quantum error correction.

Problem

• quantum error correction
• message-passing decoding
• scalability
• interpretability
• quaternary

Innovation

• dilution method
• quaternary message-passing
• Min-Sum decoding
• quantum error correction
• scalable decoding