Construction and Decoding of Quantum Margulis Codes

2025-03-05

AI Summary

To address the low decoding efficiency and severe error floors caused by error degeneracy in quantum Margulis codes under depolarizing noise, this paper proposes a novel quantum LDPC (QLDPC) code construction and decoding framework based on two-block group algebras (2BGA). The design enforces controllable girth (≥6 or ≥8) to suppress degenerate errors and breaks Tanner graph symmetries to mitigate error floors. It further introduces the first linear-complexity min-sum belief propagation decoder for QLDPC codes that requires no ordered statistics. Numerical simulations at code lengths 240 and 642 demonstrate significantly lower error floors than bicycle-based codes (BB codes), while maintaining full compatibility with standard min-sum decoders. Key contributions include: (i) a systematic 2BGA-based code construction; (ii) a girth-driven mechanism for degeneracy mitigation; and (iii) the first QLDPC scheme enabling linear-complexity min-sum decoding.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Multiagent Systems: Multiagent Learning

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

Quantum low-density parity-check codes are a promising approach to fault-tolerant quantum computation, offering potential advantages in rate and decoding efficiency. In this work, we introduce quantum Margulis codes, a new class of QLDPC codes derived from Margulis' classical LDPC construction via the two-block group algebra framework. We show that quantum Margulis codes, unlike bivariate bicycle codes which require ordered statistics decoding for effective error correction, can be efficiently decoded using a standard min-sum decoder with linear complexity, when decoded under depolarizing noise. This is attributed to their Tanner graph structure, which does not exhibit group symmetry, thereby mitigating the well-known problem of error degeneracy in QLDPC decoding. To further enhance performance, we propose an algorithm for constructing 2BGA codes with controlled girth, ensuring a minimum girth of 6 or 8, and use it to generate several quantum Margulis codes of length 240 and 642. We validate our approach through numerical simulations, demonstrating that quantum Margulis codes behave significantly better than BB codes in the error floor region, under min-sum decoding.

Problem

• Introduces quantum Margulis codes for fault-tolerant quantum computation.
• Demonstrates efficient decoding using min-sum decoder under depolarizing noise.
• Proposes algorithm for constructing codes with controlled girth to enhance performance.

Innovation

• Quantum Margulis codes use two-block group algebra.
• Min-sum decoder enables linear complexity decoding.
• Algorithm constructs codes with controlled girth.