Design and Analysis of Quantum Dual-Containing CSS LDPC Codes based on Quasi-Dyadic Matrices

2026-05-05

AI Summary

This work addresses the pressing need for efficient quantum error-correcting codes in scalable quantum computing by proposing two families of high-rate quantum dual-containing (DC) CSS LDPC codes constructed from quasi-binary matrices. While preserving the dual-containing structure necessary for transversal Hadamard gates, these codes are optimized through careful analysis of cyclic properties, automorphism groups, and minimum distance estimates to enhance finite-length error correction performance. A low-complexity binary belief propagation decoder is employed for efficient decoding. Numerical simulations demonstrate that the proposed constructions significantly outperform existing DC codes across a range of code lengths and rates in terms of finite-length block error rate, offering a promising avenue toward practical fault-tolerant quantum computation.

Abstract

Building scalable quantum computers requires quantum error-correcting codes that enable reliable operations in the presence of noise. Motivated by such need, this paper introduces two constructions of high-rate, quantum dual-containing (DC) Calderbank-Shor-Steane (CSS) low-density parity-check (LDPC) codes based on quasi-dyadic matrices. Their DC structure enables the transversal implementation of the Hadamard gate, and, jointly with the sparsity of their parity-check matrices enable low-complexity decoding via a standard binary belief-propagation algorithm. We provide several theoretical results concerning the cycle properties of these CSS codes. We also investigate their automorphism groups as well as their minimum distance. Furthermore, through numerical simulations, we show that the quantum CSS LDPC codes obtained through these constructions achieve better finite-length error rate performance than existing DC codes across different block lengths and code rates.

Problem

• quantum error-correcting codes
• dual-containing CSS codes
• LDPC codes
• quasi-dyadic matrices
• scalable quantum computing

Innovation

• quantum dual-containing codes
• CSS LDPC codes
• quasi-dyadic matrices
• transversal Hadamard gate
• belief-propagation decoding