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Quantum Error Correction with Girth-16 Non-Binary LDPC Codes via Affine Permutation Construction

Kenta Kasai

TL;DR

This paper tackles the challenge of scaling quantum error correction by increasing the Tanner graph girth of non-binary LDPC-based CSS codes from the conventional limit of $12$ to $16$ using affine permutation matrices. It introduces a randomized sequential construction for sequences of affine permutations to build $C_X$ and $C_Z$ that avoid short cycles, enabling a Code with girth $16$. Numerical results show a substantial reduction in low-weight codewords and a stronger upper bound on the minimum distance ($d_{min} \le 14$) with a notably improved error floor, albeit with a slight degradation in the waterfall region. The approach offers a viable path to more reliable, large-scale quantum error correction and points to extensions such as larger $L$, spatial coupling, and analytic error-floor modeling.

Abstract

We propose a method for constructing quantum error-correcting codes based on non-binary low-density parity-check codes with Tanner graph girth 16. While conventional constructions using circulant permutation matrices are limited to girth 12, our method employs affine permutation matrices and a randomized sequential selection procedure to eliminate short cycles and achieve girth 16. Numerical experiments show that the proposed codes significantly reduce the number of low-weight codewords. Joint belief propagation decoding over depolarizing channels reveals that although a slight degradation appears in the waterfall region, a substantial improvement is achieved in the error floor performance. We also evaluated the minimum distance and found that the proposed codes achieve a larger upper bound compared to conventional constructions.

Quantum Error Correction with Girth-16 Non-Binary LDPC Codes via Affine Permutation Construction

TL;DR

This paper tackles the challenge of scaling quantum error correction by increasing the Tanner graph girth of non-binary LDPC-based CSS codes from the conventional limit of to using affine permutation matrices. It introduces a randomized sequential construction for sequences of affine permutations to build and that avoid short cycles, enabling a Code with girth . Numerical results show a substantial reduction in low-weight codewords and a stronger upper bound on the minimum distance () with a notably improved error floor, albeit with a slight degradation in the waterfall region. The approach offers a viable path to more reliable, large-scale quantum error correction and points to extensions such as larger , spatial coupling, and analytic error-floor modeling.

Abstract

We propose a method for constructing quantum error-correcting codes based on non-binary low-density parity-check codes with Tanner graph girth 16. While conventional constructions using circulant permutation matrices are limited to girth 12, our method employs affine permutation matrices and a randomized sequential selection procedure to eliminate short cycles and achieve girth 16. Numerical experiments show that the proposed codes significantly reduce the number of low-weight codewords. Joint belief propagation decoding over depolarizing channels reveals that although a slight degradation appears in the waterfall region, a substantial improvement is achieved in the error floor performance. We also evaluated the minimum distance and found that the proposed codes achieve a larger upper bound compared to conventional constructions.

Paper Structure

This paper contains 11 sections, 1 theorem, 19 equations, 1 figure, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Permutation Matrices for LDPC Codes
  3. Permutation Matrices
  4. Affine and Circulant Permutation Matrices
  5. Low-Density Parity-Check Matrices
  6. Block Cycles
  7. Komoto--Kasai Construction
  8. Proposed Construction Method of $C_X,C_Z$
  9. Numerical Results
  10. Computing Upper Bound of Minimum Distance
  11. Conclusion and Future Work

Key Result

Theorem 1

Let $a, b, c, d, e, f \in \mathcal{F}_P$ be mutually commuting elements. In particular, this commutativity condition is automatically satisfied when $a, b, c, d, e, f \in \mathcal{Q}_P$. Consider the following $2 \times 3$ subarray of a larger permutation array $H$: Then, the block cycle of length 12 formed by the sequence $a \to b \to c \to a \to b \to c \to a$ is a closed. Moreover, it forms a

Figures (1)

  • Figure 1: Frame error rate comparison between the proposed $[[806400, 403200, \le\! 14]]$ code and the conventional $[[806400, 403200, \le 9]]$ code komoto2024quantumerrorcorrectionnear.

Theorems & Definitions (5)

  • Example 1
  • Theorem 1: 1317123
  • proof
  • Definition 1
  • Example 2