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Learning to Decode Quantum LDPC Codes Via Belief Propagation

Mohsen Moradi, Vahid Nourozi, Salman Habib, David G. M. Mitchell

TL;DR

Simulation results demonstrate the superiority of the proposed RL-based QLDPC decoders in terms of performance and convergence speed when compared to flooding and random sequential schedules, while achieving performance competitive with state-of-the-art BP-based decoders at comparable complexity.

Abstract

Belief-propagation (BP) decoding for quantum low-density parity-check (QLDPC) codes is appealing due to its low complexity, yet it often exhibits convergence issues due to quantum degeneracy and short cycles that exist in the Tanner graph. To overcome this challenge, this paper proposes a reinforcement-learning (RL) approach that learns (offline) how to decode QLDPC codes based on sequential decoding trajectories. The decoding is formulated as a Markov decision process with a local, syndrome-driven state representation of the underlying RL agent. To enable fast inference, critical for practical implementation, we incrementally update our RL-based QLDPC decoder using second-order neighborhoods that avoid global rescans. Simulation results on representative QLDPC codes demonstrate the superiority of the proposed RL-based QLDPC decoders in terms of performance and convergence speed when compared to flooding and random sequential schedules, while achieving performance competitive with state-of-the-art BP-based decoders at comparable complexity.

Learning to Decode Quantum LDPC Codes Via Belief Propagation

TL;DR

Simulation results demonstrate the superiority of the proposed RL-based QLDPC decoders in terms of performance and convergence speed when compared to flooding and random sequential schedules, while achieving performance competitive with state-of-the-art BP-based decoders at comparable complexity.

Abstract

Belief-propagation (BP) decoding for quantum low-density parity-check (QLDPC) codes is appealing due to its low complexity, yet it often exhibits convergence issues due to quantum degeneracy and short cycles that exist in the Tanner graph. To overcome this challenge, this paper proposes a reinforcement-learning (RL) approach that learns (offline) how to decode QLDPC codes based on sequential decoding trajectories. The decoding is formulated as a Markov decision process with a local, syndrome-driven state representation of the underlying RL agent. To enable fast inference, critical for practical implementation, we incrementally update our RL-based QLDPC decoder using second-order neighborhoods that avoid global rescans. Simulation results on representative QLDPC codes demonstrate the superiority of the proposed RL-based QLDPC decoders in terms of performance and convergence speed when compared to flooding and random sequential schedules, while achieving performance competitive with state-of-the-art BP-based decoders at comparable complexity.
Paper Structure (28 sections, 1 theorem, 87 equations, 7 figures, 4 tables, 1 algorithm)

This paper contains 28 sections, 1 theorem, 87 equations, 7 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Stabilizer and CSS QLDPC Codes
  4. Syndrome decoding and degeneracy (X-only channel)
  5. Residual Mismatch
  6. Sequential BP Decoding
  7. Motivation for RL-based Scheduling
  8. RL Formulation for Scheduling VN Updates
  9. Fast Inference and Fast Monte-Carlo Evaluation
  10. Edge Indexing and Adjacency Arrays
  11. Incremental Maintenance of Local States $\sigma_i$
  12. SVNS Check-products (Cached $\tanh$-products)
  13. Fast Update After a Hard-decision Flip (Residual, States, and Priorities)
  14. Greedy Selection with a Max-heap (Priority Queue)
  15. Main Inference Loop
  16. ...and 13 more sections

Key Result

Lemma 1

For a measured syndrome $\bm{s}$, and the current hard estimate $\hat{\bm{e}}$ with the residual mismatch vector $\bm{\delta} = \bm{s} \oplus (H_1 \hat{\bm{e}}),$ if we flip the $i$-th bit of the hard estimate $\hat{\bm{e}}$ then the updated residual mismatch bits flip exactly on the neighboring che

Figures (7)

  • Figure 1: FER for the $[[882, 24, 18\leq d \leq 24]]$ code B1 with flooding BP and SVNS-BP vs. our proposed RL-SVNS-BP decoding algorithm.
  • Figure 2: FER for the $[[882,48,16]]$ code B2 over the depolarizing channel.
  • Figure 3: FER for the $[[180,10,15 \leq d \leq 18]]$ code A5 over the depolarizing channel.
  • Figure 4: FER for the $[[144,12,12]]$ BB code over the depolarizing channel.
  • Figure 5: FER for the $[[288,12,18]]$ BB code over the depolarizing channel.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Example 1
  • Example 2