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Quantum CSS LDPC Codes based on Dyadic Matrices for Belief Propagation-based Decoding

Alessio Baldelli, Massimo Battaglioni, Jonathan Mandelbaum, Sisi Miao, Laurent Schmalen

TL;DR

This work addresses designing quantum LDPC codes that balance high error-correction capability with practical decoding complexity. It introduces a dyadic-matrix–based algebraic framework to construct both classical QD-LDPC and CSS QLDPC codes, guaranteeing Tanner graphs with girth $6$ for the classical part and CAMEL-compatible structure for the quantum part. The authors show how affine exponent matrices and DPM lifting yield parity-check matrices that concentrate short cycles around a single node and satisfy the CAMEL condition, enabling effective quaternary belief-propagation decoding. Numerical results on depolarizing channels demonstrate that CAMEL decoding mitigates error floors and delivers performance competitive with or superior to several state-of-the-art codes, particularly for higher-rate designs. The approach offers scalable, high-rate constructions with practical decoding suitable for fault-tolerant quantum computation.

Abstract

Quantum low-density parity-check (QLDPC) codes provide a practical balance between error-correction capability and implementation complexity in quantum error correction (QEC). In this paper, we propose an algebraic construction based on dyadic matrices for designing both classical and quantum LDPC codes. The method first generates classical binary quasi-dyadic LDPC codes whose Tanner graphs have girth 6. It is then extended to the Calderbank-Shor-Steane (CSS) framework, where the two component parity-check matrices are built to satisfy the compatibility condition required by the recently introduced CAMEL-ensemble quaternary belief propagation decoder. This compatibility condition ensures that all unavoidable cycles of length 4 are assembled in a single variable node, allowing the mitigation of their detrimental effects by decimating that variable node.

Quantum CSS LDPC Codes based on Dyadic Matrices for Belief Propagation-based Decoding

TL;DR

This work addresses designing quantum LDPC codes that balance high error-correction capability with practical decoding complexity. It introduces a dyadic-matrix–based algebraic framework to construct both classical QD-LDPC and CSS QLDPC codes, guaranteeing Tanner graphs with girth for the classical part and CAMEL-compatible structure for the quantum part. The authors show how affine exponent matrices and DPM lifting yield parity-check matrices that concentrate short cycles around a single node and satisfy the CAMEL condition, enabling effective quaternary belief-propagation decoding. Numerical results on depolarizing channels demonstrate that CAMEL decoding mitigates error floors and delivers performance competitive with or superior to several state-of-the-art codes, particularly for higher-rate designs. The approach offers scalable, high-rate constructions with practical decoding suitable for fault-tolerant quantum computation.

Abstract

Quantum low-density parity-check (QLDPC) codes provide a practical balance between error-correction capability and implementation complexity in quantum error correction (QEC). In this paper, we propose an algebraic construction based on dyadic matrices for designing both classical and quantum LDPC codes. The method first generates classical binary quasi-dyadic LDPC codes whose Tanner graphs have girth 6. It is then extended to the Calderbank-Shor-Steane (CSS) framework, where the two component parity-check matrices are built to satisfy the compatibility condition required by the recently introduced CAMEL-ensemble quaternary belief propagation decoder. This compatibility condition ensures that all unavoidable cycles of length 4 are assembled in a single variable node, allowing the mitigation of their detrimental effects by decimating that variable node.
Paper Structure (10 sections, 4 theorems, 25 equations, 3 figures, 2 tables)

This paper contains 10 sections, 4 theorems, 25 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Classical and Quantum Codes
  4. CAMEL
  5. PCM based on DPM
  6. Code design
  7. Classical QD-LDPC Codes Design with Girth $6$
  8. Quantum LDPC Code Design
  9. Numerical results
  10. Conclusions

Key Result

Lemma 1

baldelli_ISTC_2025 Let $\mathbf P \in \mathbb F_{2^{\ell}}^{w\times N}$ be the exponent matrix of a QD-LDPC code. By applying the lifting procedure on $\mathbf P$ using the associated DPM of size $2^\ell\times 2^\ell$, we obtain the Tanner graph of the corresponding binary PCM $\mathbf H$ which cont with $i_1\neq i_2$ and $j_1\neq j_2$.

Figures (3)

  • Figure 1: Block diagram of the CAMEL decoder according to SisiCamel.
  • Figure 2: LER of QLDPC codes as a function of the depolarizing probability $p$.
  • Figure 3: LER of QLDPC codes as a function of the depolarizing probability $p$.

Theorems & Definitions (7)

  • Definition 1
  • Lemma 1
  • Theorem 1
  • Remark 1
  • Lemma 2
  • Theorem 2
  • Example 1