Power and Limitations of Linear Programming Decoder for Quantum LDPC Codes

2025-08-06

AI Summary

Linear programming (LP) decoding of quantum low-density parity-check (LDPC) codes suffers from a fundamental limitation: certain constant-weight error patterns induce ambiguous fractional solutions that cannot be uniquely rounded, leading to decoding failure. This work provides the first systematic characterization of this intrinsic limitation. To overcome it, we propose a novel LP-OSD joint decoding framework: LP generates an initial soft-decision output, which is then refined via ordered statistics decoding (OSD) leveraging residual confidence information. Experimental evaluation on medium-length quantum LDPC codes—spanning hundreds of qubits—demonstrates that our approach significantly outperforms belief propagation (BP) decoding with identical OSD post-processing. It thus breaks the performance ceiling of conventional LP decoding, offering an efficient and robust pathway toward practical quantum error correction.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

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.

Problem

• Exploring performance and limitations of LP decoders for quantum LDPC codes
• Addressing ambiguous fractional solutions in LP decoding via OSD post-processing
• Enhancing LP decoding to outperform belief propagation for intermediate-scale quantum codes

Innovation

• Linear programming decoder for quantum LDPC codes
• Post-processing with ordered statistics decoding (OSD)
• Enhanced performance for intermediate quantum code sizes