Random Construction of Quantum LDPC Codes

TL;DR

This paper introduces a scalable method for creating randomized quantum LDPC codes by locally modifying orthogonal sparse matrices, enhancing their structure for improved decoding performance.

Contribution

It presents a novel local modification technique involving cross-swaps and integer-linear-program repairs that generate ensembles of randomized quantum LDPC codes while preserving key properties.

Findings

The method maintains row and column weight distributions.

It ensures scalability with complexity depending only on maximum weights.

Enables construction of large, randomized quantum LDPC code ensembles.

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 $2\times2$ 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.