An almost-linear time decoding algorithm for quantum LDPC codes under circuit-level noise

TL;DR

This paper presents an almost-linear time decoding algorithm for quantum LDPC codes that improves decoding efficiency under circuit-level noise by combining belief propagation with a novel graph sparsification technique.

Contribution

The authors introduce the BP+OTF decoding algorithm with a graph sparsification method, enabling fast and effective decoding of quantum LDPC codes under realistic noise conditions.

Findings

Achieves similar error suppression as state-of-the-art decoders.

Maintains almost-linear runtime complexity.

Effective under circuit-level noise with sparsified error models.

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.