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Efficient Mitigation of Error Floors in Quantum Error Correction using Non-Binary Low-Density Parity-Check Codes

Kenta Kasai

TL;DR

This work tackles the stubborn error floor in quantum error correction using non-binary LDPC codes by exploiting cycle structures in the parity-check matrix. It introduces a three-type taxonomy (Type I–III) for cycles where noise estimation stalls and develops tailored, cycle-aware decoding steps whose complexity does not depend on code length. The method yields substantial reductions in the error floor, especially for the challenging $L=6$ case, though certain Type-II cycles appear intrinsic to the code construction and would require redesigned codes to fully eliminate. Overall, the approach advances practical quantum LDPC decoding by marrying finite-field cycle analysis with targeted postprocessing, suggesting paths toward deeper floors via code design changes.

Abstract

In this paper, we propose an efficient method to reduce error floors in quantum error correction using non-binary low-density parity-check (LDPC) codes. We identify and classify cycle structures in the parity-check matrix where estimated noise becomes trapped, and develop tailored decoding methods for each cycle type. For Type-I cycles, we propose a method to make the difference between estimated and true noise degenerate. Type-II cycles are shown to be uncorrectable, while for Type-III cycles, we utilize the fact that cycles in non-binary LDPC codes do not necessarily correspond to codewords, allowing us to estimate the true noise. Our method significantly improves decoding performance and reduces error floors.

Efficient Mitigation of Error Floors in Quantum Error Correction using Non-Binary Low-Density Parity-Check Codes

TL;DR

This work tackles the stubborn error floor in quantum error correction using non-binary LDPC codes by exploiting cycle structures in the parity-check matrix. It introduces a three-type taxonomy (Type I–III) for cycles where noise estimation stalls and develops tailored, cycle-aware decoding steps whose complexity does not depend on code length. The method yields substantial reductions in the error floor, especially for the challenging case, though certain Type-II cycles appear intrinsic to the code construction and would require redesigned codes to fully eliminate. Overall, the approach advances practical quantum LDPC decoding by marrying finite-field cycle analysis with targeted postprocessing, suggesting paths toward deeper floors via code design changes.

Abstract

In this paper, we propose an efficient method to reduce error floors in quantum error correction using non-binary low-density parity-check (LDPC) codes. We identify and classify cycle structures in the parity-check matrix where estimated noise becomes trapped, and develop tailored decoding methods for each cycle type. For Type-I cycles, we propose a method to make the difference between estimated and true noise degenerate. Type-II cycles are shown to be uncorrectable, while for Type-III cycles, we utilize the fact that cycles in non-binary LDPC codes do not necessarily correspond to codewords, allowing us to estimate the true noise. Our method significantly improves decoding performance and reduces error floors.
Paper Structure (12 sections, 1 theorem, 22 equations, 1 figure, 1 algorithm)

This paper contains 12 sections, 1 theorem, 22 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preparation
  3. Code Construction
  4. Conventional Noise Estimation Method
  5. Classification of Noise Estimation Errors Causing the Error Floor
  6. Cycle Identification Method
  7. Proposed Decoding Method
  8. Type-I
  9. Type-II
  10. Type-III
  11. Numerical Experiments
  12. Conclusion and Future Work

Key Result

Theorem 1

For $s_i, t_i, x_j, z_j \in \mathbb{F}_2^e$, we define $\sigma_i, \tau_i, \xi_j, \zeta_j \in \mathbb{F}_q$ are such that $\underline{w}\left(\sigma_i\right)=s_i, \underline{v}\left(\tau_i\right)=t_i, \underline{w}\left(\xi_j\right)=$$x_j$ and $\underline{v}\left(\zeta_j\right)=z_j$ hold. Note that $ where $H_\Gamma^{(i)}$ and $H_\Delta^{(j)}$ are the $i$-th and $j$-th row vectors of $H_\Gamma$ and

Figures (1)

  • Figure 1: Decoding performance $(f_m, \mathrm{FER})$ of the proposed method (dashed).

Theorems & Definitions (2)

  • Theorem 1: Finite field representation of $\hat{E}^\dagger E\in S$
  • proof