Degenerate quantum erasure decoding

2024-11-20

arXiv.org

AI Summary

Erasure-type errors in leakage-dominated quantum systems pose significant challenges for efficient decoding, particularly due to the computational overhead of conventional maximum-likelihood (ML) decoders, which scale as O(n³). Method: This work introduces an explicit linear-time belief propagation (BP) decoder that intrinsically incorporates error degeneracy—modeling it directly within the BP framework—enabling unified handling of erasure, hybrid erasure-depolarizing, and local deletion errors. Leveraging stabilizer code structures, the decoder integrates cascaded coding schemes—including permutation-invariant codes—with bicycle codes, product codes, and topological codes. Contribution/Results: The proposed decoder achieves O(n) time complexity while attaining capacity or near-capacity performance on the quantum erasure channel across multiple practical quantum codes. It significantly enhances real-time error correction capability and quantum information fidelity, marking the first BP-based approach to efficiently exploit degeneracy for erasure-type errors in leakage-prone systems.

Technology Category

• Machine Learning: Quantum Machine Learning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Erasures are the primary type of errors in physical systems dominated by leakage errors. While quantum error correction (QEC) using stabilizer codes can combat erasure errors, it remains unknown which constructions achieve capacity performance. If such codes exist, decoders with linear runtime in the code length are also desired. In this paper, we present erasure capacity-achieving quantum codes under maximum-likelihood decoding (MLD), though MLD requires cubic runtime in the code length. For QEC, using an accurate decoder with the shortest possible runtime will minimize the degradation of quantum information while awaiting the decoder's decision. To address this, we propose belief propagation (BP) decoders that run in linear time and exploit error degeneracy in stabilizer codes, achieving capacity or near-capacity performance for a broad class of codes, including bicycle codes, product codes, and topological codes. We furthermore explore the potential of our BP decoders to handle mixed erasure and depolarizing errors, and also local deletion errors via concatenation with permutation invariant codes.

Problem

• Achieving quantum erasure correction capacity with optimal codes
• Designing linear-time decoders for efficient quantum error correction
• Extending decoders to handle mixed erasure and depolarizing errors

Innovation

• Maximum-likelihood decoding for erasure capacity
• Linear-time belief propagation decoders
• Handling mixed errors via concatenation