Random Construction of Quantum LDPC Codes

2025-11-06

AI Summary

This work addresses the lack of structural randomness in orthogonal sparse matrix pairs for CSS-type quantum LDPC code construction, which limits decoding performance. We propose a randomized construction method that preserves row and column weight distributions. Our approach features: (1) localized structural perturbation via 2×2 cross-swap operations, and (2) orthogonality-preserving local repair using integer linear programming. The method efficiently generates diverse code ensembles while strictly maintaining both sparsity and orthogonality; its repair complexity depends only on the maximum row/column weight—not on matrix size. Experimental results demonstrate that the resulting quantum LDPC code ensembles significantly improve belief propagation (BP) decoding performance and exhibit excellent scalability, making them suitable for large-scale fault-tolerant quantum computing systems.

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: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We propose a method for modifying orthogonal sparse matrix pairs used in CSS codes while preserving their matrix row and column weight distributions, which play a crucial role in determining the performance of belief-propagation decoding. Unlike simple row or column permutations that merely reorder existing elements, the proposed local modification introduces genuine structural randomness through small $2 imes2$ cross-swap operations followed by integer-linear-program-based local repairs that restore orthogonality. By applying this procedure repeatedly in a random manner, ensembles of randomized quantum LDPC codes can be constructed. The computational complexity of each repair depends only on the maximum row and column weights and is independent of the overall matrix size, ensuring scalability to large code blocks.

Problem

• Modifying CSS code matrices while preserving weight distributions for decoding performance
• Introducing structural randomness via cross-swap operations and local orthogonality repairs
• Creating scalable quantum LDPC code ensembles with size-independent computational complexity

Innovation

• Random cross-swap operations modify matrix pairs
• Integer-linear-program repairs restore orthogonality locally
• Scalable complexity independent of overall matrix size