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Explicit Construction of Quantum Quasi-Cyclic Low-Density Parity-Check Codes with Column Weight 2 and Girth 12

Daiki Komoto, Kenta Kasai

TL;DR

The paper addresses the challenge of constructing quantum QC-LDPC codes with the maximum Tanner-graph girth of $12$ while maintaining the orthogonality required for CSS quantum codes. It delivers a deterministic, explicit construction for $(2,L,P)$ QC-LDPC matrices achieving girth $12$ and extends this to orthogonal QC-LDPC pairs suitable for quantum error correction; further, it provides finite-field extensions to non-binary NB-LDPC codes and spatially-coupled variants that preserve both orthogonality and girth. Key contributions include a rigorous classical construction with $J=2$, an orthogonal matrix-pair framework following Hagiwara–Imai, and explicit conditions for finite-field extension and spatial coupling, supported by numerical and theoretical analysis. The results offer a scalable, hardware-friendly pathway to high-performance quantum LDPC codes without resorting to random search, with demonstrated applicability to NB-LDPC and SC-LDPC architectures. This work thus advances practical fault-tolerant quantum communication and scalable quantum computation by delivering guaranteed girth and orthogonality in structured quantum error-correcting codes.

Abstract

This study proposes an explicit construction method for quantum quasi-cyclic low-density parity-check (QC-LDPC) codes with a girth of 12. The proposed method designs parity-check matrices that maximize the girth while maintaining an orthogonal structure suitable for quantum error correction. By utilizing algebraic techniques, short cycles are eliminated, which improves error correction performance. Additionally, this method is extended to non-binary LDPC codes and spatially-coupled LDPC codes, demonstrating that both the girth and orthogonality can be preserved. The results of this study enable the design of high-performance quantum error-correcting codes without the need for random search.

Explicit Construction of Quantum Quasi-Cyclic Low-Density Parity-Check Codes with Column Weight 2 and Girth 12

TL;DR

The paper addresses the challenge of constructing quantum QC-LDPC codes with the maximum Tanner-graph girth of while maintaining the orthogonality required for CSS quantum codes. It delivers a deterministic, explicit construction for QC-LDPC matrices achieving girth and extends this to orthogonal QC-LDPC pairs suitable for quantum error correction; further, it provides finite-field extensions to non-binary NB-LDPC codes and spatially-coupled variants that preserve both orthogonality and girth. Key contributions include a rigorous classical construction with , an orthogonal matrix-pair framework following Hagiwara–Imai, and explicit conditions for finite-field extension and spatial coupling, supported by numerical and theoretical analysis. The results offer a scalable, hardware-friendly pathway to high-performance quantum LDPC codes without resorting to random search, with demonstrated applicability to NB-LDPC and SC-LDPC architectures. This work thus advances practical fault-tolerant quantum communication and scalable quantum computation by delivering guaranteed girth and orthogonality in structured quantum error-correcting codes.

Abstract

This study proposes an explicit construction method for quantum quasi-cyclic low-density parity-check (QC-LDPC) codes with a girth of 12. The proposed method designs parity-check matrices that maximize the girth while maintaining an orthogonal structure suitable for quantum error correction. By utilizing algebraic techniques, short cycles are eliminated, which improves error correction performance. Additionally, this method is extended to non-binary LDPC codes and spatially-coupled LDPC codes, demonstrating that both the girth and orthogonality can be preserved. The results of this study enable the design of high-performance quantum error-correcting codes without the need for random search.
Paper Structure (14 sections, 13 theorems, 30 equations, 1 figure, 1 table)

This paper contains 14 sections, 13 theorems, 30 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. QC-LDPC Matrices
  3. Cycles in the LDPC Matrix
  4. Orthogonal QC-LDPC Matrix Pairs with Girth 12 for Quantum Error Correction
  5. Explicit Construction of Classical Codes
  6. Explicit Construction of Orthogonal QC-LDPC Matrix Pairs
  7. Finite Field Extension
  8. Numerical Results
  9. Spatial Coupling
  10. Conclusion
  11. Proofs in Section \ref{['sec:Explicit Construction of Classical QC-LDPC Matrices with Girth 12']}
  12. Proofs in Section \ref{['sec:Orthogonal QC-LDPC Matrix Pairs with Girth 12 for Quantum Error Correction']}
  13. Proofs in Section \ref{['sec:finite-field-extension']}
  14. Proofs in Section \ref{['sec:spatial-coupling']}

Key Result

Theorem 3

For any $L \geq 6$ and $P \geq P_0 \overset{\text{\small def}}{=} 2^{L+1}$, the $(2, L, P)$-QC-LDPC matrix $\hat{H}$ constructed according to Definition dfn:classicalQC-LDPC-matrix has a girth of 12.

Figures (1)

  • Figure 1: Example of $F(\hat{H}_X)$ and $F(\hat{H}_Z)$ for $L = 6$, $P = 49$.

Theorems & Definitions (30)

  • Definition 1: CPM and QC-LDPC Matrices
  • Definition 2: Construction of Classical QC-LDPC Matrices
  • Theorem 3
  • Proposition 4
  • Definition 5: Construction of Orthogonal QC-LDPC Matrix Pairs
  • Theorem 6: Graph Isomorphism of Orthogonal QC-LDPC Matrix Pairs, Generalization of hagiwara2007quantum
  • Theorem 7
  • Theorem 8: Conditions for Finite Field Extension, komoto2024quantum
  • Theorem 9
  • Definition 10: Construction of Spatially-Coupled QC-LDPC Matrix Pairs
  • ...and 20 more