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

TL;DR

This paper presents novel high-rate quantum CSS LDPC codes based on quasi-dyadic matrices, enabling efficient decoding and transversal Hadamard implementation, with demonstrated improved error performance over existing codes.

Contribution

Introduces two constructions of quantum dual-containing CSS LDPC codes using quasi-dyadic matrices, enhancing decoding efficiency and error performance.

Findings

Codes enable transversal Hadamard gate implementation.

Codes achieve better finite-length error rates than existing DC codes.

Theoretical analysis of cycle properties, automorphisms, and minimum distance.

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.