Logical Neural Belief Propagation for Linear-Complexity Decoding of Surface Codes
Abstract
Quantum error correction (QEC) requires accurate and efficient decoders, yet belief propagation (BP), despite its linear decoding complexity, often provides insufficient logical accuracy on surface codes. We propose Logical Neural Belief Propagation (L-NBP), a BP-based neural decoder that redirects the decoding objective from physical-level to logical-level decoding. L-NBP uses a neural BP (NBP) module to produce posterior beliefs, which a logical classifier transforms into a continuous-valued soft syndrome for logical-operator prediction. Trained end-to-end by backpropagation, the NBP module learns soft syndromes that are favorable for logical classification. On surface codes, L-NBP matches or outperforms BP with ordered-statistics decoding (BP-OSD) and minimum-weight perfect matching (MWPM) while retaining the linear complexity of BP, and achieves a threshold of 17.5% under depolarizing noise. Under circuit-level noise, L-NBP matches the accuracy of BP-OSD on the distance-$9$ surface code while requiring only 0.2% of its complexity.
Related
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BibTeX
@misc{kwak2026logical,
title = {{Logical Neural Belief Propagation for Linear-Complexity Decoding of Surface Codes}},
author = {Hee-Youl Kwak and Seong-Joon Park and Dae-Young Yun and Eliya Nachmani and Jae-Won Kim},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.27682},
eprint = {2608.27682},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27682},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Logical Neural Belief Propagation for Linear-Complexity Decoding of Surface Codes
AU - Hee-Youl Kwak
AU - Seong-Joon Park
AU - Dae-Young Yun
AU - Eliya Nachmani
AU - Jae-Won Kim
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2608.27682
UR - https://arxiv.org/abs/2608.27682
AB - Quantum error correction (QEC) requires accurate and efficient decoders, yet belief propagation (BP), despite its linear decoding complexity, often provides insufficient logical accuracy on surface codes. We propose Logical Neural Belief Propagation (L-NBP), a BP-based neural decoder that redirects the decoding objective from physical-level to logical-level decoding. L-NBP uses a neural BP (NBP) module to produce posterior beliefs, which a logical classifier transforms into a continuous-valued soft syndrome for logical-operator prediction. Trained end-to-end by backpropagation, the NBP module learns soft syndromes that are favorable for logical classification. On surface codes, L-NBP matches or outperforms BP with ordered-statistics decoding (BP-OSD) and minimum-weight perfect matching (MWPM) while retaining the linear complexity of BP, and achieves a threshold of 17.5% under depolarizing noise. Under circuit-level noise, L-NBP matches the accuracy of BP-OSD on the distance-$9$ surface code while requiring only 0.2% of its complexity.
ER -