An almost-linear time decoding algorithm for quantum LDPC codes under circuit-level noise
Abstract
Fault-tolerant quantum computers must be designed in conjunction with classical co-processors that decode quantum error correction measurement information in real-time. In this work, we introduce the belief propagation plus ordered Tanner forest (BP+OTF) algorithm as an almost-linear time decoder for quantum low-density parity-check codes. The OTF post-processing stage removes qubits from the decoding graph until it has a tree-like structure. Provided that the resultant loop-free OTF graph supports a subset of qubits that can generate the syndrome, BP decoding is then guaranteed to converge. To enhance performance under circuit-level noise, we introduce a technique for sparsifying detector error models. This method uses a transfer matrix to map soft information from the full detector graph to the sparsified graph, preserving critical error propagation information from the syndrome extraction circuit. Our BP+OTF implementation first applies standard BP to the full detector graph, followed by BP+OTF post-processing on the sparsified graph. Numerical simulations show that the BP+OTF decoder achieves similar logical error suppression compared to state-of-the-art inversion-based and matching decoders for bivariate bicycle and surface codes, respectively, while maintaining almost-linear runtime complexity across all stages.
Keywords
Cite
@article{arxiv.2409.01440,
title = {An almost-linear time decoding algorithm for quantum LDPC codes under circuit-level noise},
author = {Antonio deMarti iOlius and Imanol Etxezarreta Martinez and Joschka Roffe and Josu Etxezarreta Martinez},
journal= {arXiv preprint arXiv:2409.01440},
year = {2025}
}
Comments
18 pages, 8 figures. In the V2 we include updated results for the bivariate bicycle codes, a timing analysis for those, surface code performances and a detailed analysis of the sparsification routine