Power and Limitations of Linear Programming Decoder for Quantum LDPC Codes

TL;DR

This paper investigates the capabilities and limitations of linear programming decoders for quantum LDPC codes, revealing a key ambiguity issue and proposing an effective post-processing method that improves decoding performance.

Contribution

It identifies a fundamental limitation of LP decoding for quantum codes and demonstrates that combining LP with ordered statistics decoding enhances performance for practical quantum LDPC codes.

Findings

LP decoding faces ambiguity with certain error patterns

Ordered statistics decoding improves LP decoding results

LP+OSD outperforms belief propagation for small to medium codes

Abstract

Decoding quantum error-correcting codes is a key challenge in enabling fault-tolerant quantum computation. In the classical setting, linear programming (LP) decoders offer provable performance guarantees and can leverage fast practical optimization algorithms. Although LP decoders have been proposed for quantum codes, their performance and limitations remain relatively underexplored. In this work, we uncover a key limitation of LP decoding for quantum low-density parity-check (LDPC) codes: certain constant-weight error patterns lead to ambiguous fractional solutions that cannot be resolved through independent rounding. To address this issue, we incorporate a post-processing technique known as ordered statistics decoding (OSD), which significantly enhances LP decoding performance in practice. Our results show that LP decoding, when augmented with OSD, can outperform belief propagation with the same post-processing for intermediate code sizes of up to hundreds of qubits. These findings suggest that LP-based decoders, equipped with effective post-processing, offer a promising approach for decoding near-term quantum LDPC codes.