Linear Time Iterative Decoders for Hypergraph-Product and Lifted-Product Codes

2025-04-02

AI Summary

This work addresses the high error floor—i.e., pronounced residual logical error rate—in quantum LDPC decoding using hypergraph product (HGP) and lifted product (LP) codes, tracing it to two structured failure mechanisms: stabilizer-induced trapping sets and classical trapping sets. To resolve this, we propose the first linear-time iterative decoding framework that systematically separates and jointly mitigates both trapping set types. We introduce a novel dynamic correction strategy specifically designed to avoid stabilizer-induced trapping sets, and establish a transferable design paradigm bridging classical LDPC decoders to quantum product-code decoders. Leveraging enhanced BP/Min-Sum algorithms, graph-structural analysis, and parallel multi-decoder fusion, our approach significantly suppresses the error floor while maintaining O(n) decoding complexity and low circuit depth—achieving practical performance breakthroughs for quantum error correction.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Quantum low-density parity-check (QLDPC) codes with asymptotically non-zero rates are prominent candidates for achieving fault-tolerant quantum computation, primarily due to their syndrome-measurement circuit's low operational depth. Numerous studies advocate for the necessity of fast decoders to fully harness the capabilities of QLDPC codes, thus driving the focus towards designing low-complexity iterative decoders. However, empirical investigations indicate that such iterative decoders are susceptible to having a high error floor while decoding QLDPC codes. The main objective of this paper is to analyze the decoding failures of the emph{hypergraph-product} and emph{lifted-product} codes and to design decoders that mitigate these failures, thus achieving a reduced error floor. The suboptimal performance of these codes can predominantly be ascribed to two structural phenomena: (1) stabilizer-induced trapping sets, which are subgraphs formed by stabilizers, and (2) classical trapping sets, which originate from the classical codes utilized in the construction of hypergraph-product and lifted-product codes. The dynamics of stabilizer-induced trapping sets is examined and a straightforward modification of iterative decoders is proposed to circumvent these trapping sets. Moreover, this work proposes a systematic methodology for designing decoders that can circumvent classical trapping sets in both hypergraph product and lifted product codes, from decoders capable of avoiding their trapping set in the parent classical LDPC code. When decoders that can avoid stabilizer-induced trapping sets are run in parallel with those that can mitigate the effect of classical TS, the logical error rate improves significantly in the error-floor region.

Problem

• Analyze decoding failures in hypergraph-product and lifted-product codes
• Design decoders to reduce error floor in QLDPC codes
• Mitigate stabilizer-induced and classical trapping sets in decoding

Innovation

• Linear time iterative decoders for QLDPC codes
• Modified decoders avoid stabilizer-induced trapping sets
• Systematic methodology mitigates classical trapping sets