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arXiv:2605.03631

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

Alessio Baldelli, Marco Baldi, Massimo Battaglioni, Franco Chiaraluce, Paolo Santini

cs.ITquant-phSubmitted May 5, 2026• Announced May 6, 2026• 0 citations• Revised Jul 22, 2026 (v3)
v1 · May 5, 2026v2 · May 12, 2026v3 · Jul 22, 2026
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Abstract

Quantum error correcting codes are essential to achieve fault-tolerant quantum computation. This work introduces two constructions of high-rate, dual-containing (DC) Calderbank--Shor--Steane low-density parity-check (LDPC) codes based on quasi-dyadic matrices. We characterize the automorphism group of such codes, investigate their minimum distance behavior, and provide several theoretical results on their cycle properties. Monte Carlo simulations under depolarizing and phenomenological noise show better finite-length logical error rates than the considered DC benchmark codes and competitive performance against several state-of-the-art quantum LDPC code families. Finally, we employ an automorphism-ensemble belief propagation decoder to improve their decoding performance.

Comments
16 pages, Journal paper

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We construct a new family of Calderbank-Shor-Steane (CSS) codes using the generator and parity-check matrices of Low-Density Generator Matrix (LDGM) codes, with row operations applied to both matrices in order to achieve the desired quantum rate. Decoding is performed in an iterative manner, by applying message passing over the associated graph, and discrete Density Evolution (DDE) is used to optimize performance in the depolarizing channel. The proposed construction offers high flexibility and easiness in the design, producing quantum codes that possess excellent error correction capabilities. By properly designing the structure of the code, we are able to control and bound the weight of the stabilizer generators to a small value, which results in codes particularly well suited for fault-tolerant quantum computation. At the same time, these codes achieve very good performance in terms of error correction capability.
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