Bidirectional Decoding for Concatenated Quantum Hamming Codes

TL;DR

This paper introduces a bidirectional decoding strategy for concatenated quantum Hamming codes that significantly improves error thresholds and preserves code-distance scaling, advancing fault-tolerant quantum computing.

Contribution

The work presents a polynomial-time hard-decision decoder that leverages higher-level syndrome information to enhance decoding performance of concatenated quantum Hamming codes.

Findings

Threshold improved from 1.56% to 4.35% under bit-flip noise.

Empirically preserves full code-distance scaling for multiple levels.

Achieves faster logical-error suppression than traditional local decoders.

Abstract

High-rate concatenated quantum codes offer a promising pathway toward fault-tolerant quantum computation, yet designing efficient decoders that fully exploit their error-correction capability remains a significant challenge. In this work, we introduce a hard-decision decoder for concatenated quantum Hamming codes with time complexity polynomial in the block length. This decoder overcomes the limitations of conventional local decoding by leveraging higher-level syndrome information to revise lower-level recovery decisions -- a strategy we refer to as bidirectional decoding. For the concatenated $[[15,7,3]]$ quantum Hamming code under independent bit-flip noise, the bidirectional decoder improves the threshold from approximately $1.56\%$ to $4.35\%$ compared with standard local decoding. Moreover, the decoder empirically preserves the full $3^{L}$ code-distance scaling for at least three levels of concatenation, resulting in substantially faster logical-error suppression than the $2^{L+1}$ scaling offered by local decoders. Our results can enhance the competitiveness of concatenated-code architectures for low-overhead fault-tolerant quantum computation.