An Efficient Algorithm to Sample Quantum Low-Density Parity-Check Codes

2026-01-13

AI Summary

This work addresses the efficient sampling of random sparse self-orthogonal parity-check matrices suitable for quantum low-density parity-check (LDPC) codes. The authors propose a purely combinatorial, row-by-row sampling algorithm that constructs sparse binary matrices satisfying the self-duality condition without relying on algebraic structures such as quasi-cyclic constraints. Notably, this approach introduces information set decoding (ISD) techniques into the construction of parity-check matrices—a novel application that enables generalization to non-binary fields and arbitrary quantum stabilizer LDPC codes. Theoretical analysis delineates the feasible parameter regimes and computational complexity, while numerical experiments demonstrate the algorithm’s efficiency and practicality, thereby overcoming limitations inherent in traditional algebraic constructions.

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: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

In this paper, we present an efficient algorithm to sample random sparse matrices to be used as check matrices for quantum Low-Density Parity-Check (LDPC) codes. To ease the treatment, we mainly describe our algorithm as a technique to sample a dual-containing binary LDPC code, hence, a sparse matrix $\mathbf H\in\mathbb F_2^{r\times n}$ such that $\mathbf H\mathbf H^\top = \mathbf 0$. However, as we show, the algorithm can be easily generalized to sample dual-containing LDPC codes over non binary finite fields as well as more general quantum stabilizer LDPC codes. While several constructions already exist, all of them are somewhat algebraic as they impose some specific property (e.g., the matrix being quasi-cyclic). Instead, our algorithm is purely combinatorial as we do not require anything apart from the rows of $\mathbf H$ being sparse enough. In this sense, we can think of our algorithm as a way to sample sparse, self-orthogonal matrices that are as random as possible. Our algorithm is conceptually very simple and, as a key ingredient, uses Information Set Decoding (ISD) to sample the rows of $\mathbf H$, one at a time. The use of ISD is fundamental as, without it, efficient sampling would not be feasible. We give a theoretical characterization of our algorithm, determining which ranges of parameters can be sampled as well as the expected computational complexity. Numerical simulations and benchmarks confirm the feasibility and efficiency of our approach.

Problem

• quantum LDPC codes
• sparse matrices
• dual-containing codes
• self-orthogonal matrices
• random sampling

Innovation

• quantum LDPC codes
• random sparse matrices
• self-orthogonal matrices
• Information Set Decoding
• combinatorial sampling