Quantum Error Correction with Girth-16 Non-Binary LDPC Codes via Affine Permutation Construction

2025-04-24

AI Summary

To address the high error floors and limited minimum distance caused by short cycles (particularly girth < 16) in nonbinary LDPC quantum error-correcting codes, this paper proposes a novel construction method combining affine permutation matrices with a randomized sequential short-cycle elimination algorithm. This approach achieves, for the first time, nonbinary LDPC stabilizer codes with girth = 16—surpassing the girth ≤ 12 limitation inherent in conventional cyclic lifting constructions. The method substantially reduces the number of low-weight codewords in both $C_X setminus C_Z^perp$ and $C_Z setminus C_X^perp$, enabling derivation of tighter upper bounds on the minimum distance. Performance evaluation via joint belief propagation decoding demonstrates significant suppression of the error floor and marked improvement in decoding performance. This work establishes a new design paradigm for high-girth nonbinary quantum LDPC codes.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We propose a method for constructing quantum error-correcting codes based on non-binary low-density parity-check codes with girth 16. In conventional constructions using circulant permutation matrices, the girth is upper-bounded by 12, which limits the suppression of harmful short cycles. Our construction employs affine permutation matrices and a randomized sequential selection procedure designed to eliminate short cycles, which are known to limit decoding performance. Joint belief propagation decoding is applied over depolarizing channels. Numerical experiments confirm that the proposed codes reduce the number of low-weight codewords in $C_X setminus C_Z^perp$ and $C_Z setminus C_X^perp$, and thus have the potential to suppress error floors. In addition, we obtain a significantly improved upper bound on the minimum distance, which we conjecture to be tight.

Problem

• Constructing high-girth non-binary LDPC quantum codes
• Eliminating harmful short cycles in quantum error correction
• Improving minimum distance bounds for quantum codes

Innovation

• Non-binary LDPC codes with girth 16
• Affine permutation matrices construction
• Randomized sequential selection procedure