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Low-Complexity Linear Programming Based Decoding of Quantum LDPC codes

Sana Javed, Francisco Garcia-Herrero, Bane Vasic, Mark F. Flanagan

TL;DR

This work tackles the error floor observed in belief-propagation-based decoding of quantum LDPC codes under depolarizing noise. It proposes a low-complexity syndrome-based linear programming decoder (SB-LP) and its use as post-processing after syndrome-based min-sum (SB-MS) decoding, augmented by an early stopping mechanism to reduce wasted iterations. The SB-LP employs a tailored LP relaxation with parallel per-edge updates and is enhanced when applied after SB-MS by using trapping-set awareness and initializing SB-LP from SB-MS outputs; this combination yields a 2–3 order of magnitude improvement in logical error rate on a sample code, with reduced overall iterations. The results suggest practically scalable decoding strategies for QLDPC codes that offer lower complexity and stronger error-floor suppression compared to alternatives like OSD or stabilizer inactivation.

Abstract

This paper proposes two approaches for reducing the impact of the error floor phenomenon when decoding quantum low-density parity-check codes with belief propagation based algorithms. First, a low-complexity syndrome-based linear programming (SB-LP) decoding algorithm is proposed, and second, the proposed SB-LP is applied as a post-processing step after syndrome-based min-sum (SB-MS) decoding. For the latter case, a new early stopping criterion is introduced to decide when to activate the SB-LP algorithm, avoiding executing a predefined maximum number of iterations for the SB-MS decoder. Simulation results show, for a sample hypergraph code, that the proposed decoder can lower the error floor by two to three orders of magnitude compared to SB-MS for the same total number of decoding iterations.

Low-Complexity Linear Programming Based Decoding of Quantum LDPC codes

TL;DR

This work tackles the error floor observed in belief-propagation-based decoding of quantum LDPC codes under depolarizing noise. It proposes a low-complexity syndrome-based linear programming decoder (SB-LP) and its use as post-processing after syndrome-based min-sum (SB-MS) decoding, augmented by an early stopping mechanism to reduce wasted iterations. The SB-LP employs a tailored LP relaxation with parallel per-edge updates and is enhanced when applied after SB-MS by using trapping-set awareness and initializing SB-LP from SB-MS outputs; this combination yields a 2–3 order of magnitude improvement in logical error rate on a sample code, with reduced overall iterations. The results suggest practically scalable decoding strategies for QLDPC codes that offer lower complexity and stronger error-floor suppression compared to alternatives like OSD or stabilizer inactivation.

Abstract

This paper proposes two approaches for reducing the impact of the error floor phenomenon when decoding quantum low-density parity-check codes with belief propagation based algorithms. First, a low-complexity syndrome-based linear programming (SB-LP) decoding algorithm is proposed, and second, the proposed SB-LP is applied as a post-processing step after syndrome-based min-sum (SB-MS) decoding. For the latter case, a new early stopping criterion is introduced to decide when to activate the SB-LP algorithm, avoiding executing a predefined maximum number of iterations for the SB-MS decoder. Simulation results show, for a sample hypergraph code, that the proposed decoder can lower the error floor by two to three orders of magnitude compared to SB-MS for the same total number of decoding iterations.
Paper Structure (11 sections, 2 equations, 4 figures, 3 algorithms)

This paper contains 11 sections, 2 equations, 4 figures, 3 algorithms.

Table of Contents

  1. INTRODUCTION
  2. Background
  3. Quantum Noise Model
  4. Tanner Graph
  5. Syndrome-Based Min-Sum (SB-MS) Algorithm
  6. Proposed Method
  7. Proposed Syndrome-based Linear Programming (SB-LP) Decoder
  8. SB-LP as Post-Processing after SB-MS Decoding
  9. Early Stopping Criterion for the SB-MS Decoder
  10. Simulation Results and Discussion
  11. Conclusion

Figures (4)

  • Figure 1: Evolution of the number of unmatched syndrome elements with successive decoding iterations for a sample error vector. Plots are shown for the SB-MS (blue) and SB-LP (red) decoders for the $[[882,24]]$ QLDPC code B1 from Panteleev_2021.
  • Figure 2: Performance of the SB-MS decoder, the SB-LP decoder, and the combined SB-MS and SB-LP decoder with and without early stopping criterion, for the $[[882,24]]$ QLDPC code B1 from Panteleev_2021.
  • Figure 3: Convergence behavior of the combined SB-MS and SB-LP decoder, with and without early stopping criterion, for the same sample error vector as in Fig.\ref{['fig:Ocillation']}.
  • Figure 4: Average number of iterations for the SB-MS decoder, the SB-LP decoder, and the combined SB-MS and SB-LP decoder with and without early stopping criterion, for the $[[882,24]]$ QLDPC code B1 from Panteleev_2021.