Learning-Based List Sequential Belief Propagation Decoding of Quantum LDPC Codes

Abstract: Quantum low-density parity-check (QLDPC) codes are strong candidates for fault-tolerant quantum computation, but efficient decoding remains a major challenge due to short cycles, degeneracy, and the poor convergence of standard belief-propagation (BP) decoders. We propose a reinforcement learning-based list sequential (RL-LS) BP decoder for QLDPC codes by extending the reinforcement-learning-based sequential variable-node scheduling (RL-S) framework with list-based search. At each step, the learned policy selects the next variable node to update; the decoder then retains the ordinary RL-S trajectory while also exploring a competing branch obtained by softly biasing the post-update LLR pair toward the second-most likely Pauli symbol, recomputing the incident local BP messages, and setting the visited variable node to that second-best symbol. Candidate trajectories are ranked and pruned using our proposed cumulative path metric. The resulting decoder extends the learned decoder by combining the improved convergence of learned sequential scheduling with list exploration. Numerical results on representative QLDPC benchmark codes over the depolarizing channel show that our proposed method improves the decoding performance of the underlying decoder and compares favorably with existing BP-based decoding methods.

• The paper introduces a reinforcement learning-based list sequential BP decoder that extends traditional BP with list-based path exploration to handle quantum degeneracy.
• It utilizes a cumulative path metric for adaptive pruning, achieving lower block error rates and reducing iterations on depolarizing channel errors.
• The approach supports modular, hardware-friendly implementations and scalability for practical QLDPC deployments in fault-tolerant quantum computing.

Learning-Based List Sequential Belief Propagation Decoding for Quantum LDPC Codes

Introduction

Quantum LDPC (QLDPC) codes are pivotal for scalable, low-overhead fault-tolerant quantum computing architectures. Message-passing decoding, especially belief propagation (BP), is attractive due to its local structure and low computational complexity, but suffers from degraded performance in the quantum regime because of short cycles, degeneracy, and unstable convergence properties inherent to quantum Tanner graphs. The challenge is not only to return a syndrome-consistent error estimate but to find an error in the correct logical coset, due to quantum degeneracy. Sequential scheduling and post-processing have been shown to improve BP performance, but remain vulnerable to early decision mistakes, particularly when local posteriors are ambiguous. The paper introduces a learning-based list sequential (RL-LS) BP decoder that extends RL-based variable-node scheduling with list-based path exploration to reduce decoding failure probability.

Decoding Framework and Methodology

The RL-LS decoder builds upon a reinforcement learning-based sequential BP scheduling (RL-S) paradigm, where a learned policy selects the next variable-node update based on a compact syndrome-driven local state. RL-LS generalizes RL-S by maintaining a list of decoding trajectories rather than a single trajectory. At each node update, two continuations are generated: one following the RL-S hard decision (the locally dominant Pauli symbol), and one branching toward the second-most likely symbol, with LLRs biased accordingly and subsequent BP messages recomputed. Path candidates are ranked and pruned using a cumulative metric reflecting local log-likelihood penalties from choosing non-majority symbols.

Sequential path expansion is strictly non-destructive, always preserving the RL-S trajectory, and augmenting potential solutions with locally plausible alternatives at ambiguous nodes. The list is capped at a fixed maximum size to control the complexity. The pruning criterion prioritizes paths with minimal cumulative metric and residual mismatch weight. Once any path achieves syndrome consistency, decoding halts with that candidate.

Technical Contributions

• List-based Path Expansion: RL-LS forms two local continuations at each variable node, biasing LLRs toward both the maximum and second-most probable Pauli symbols, providing path diversity in the face of local ambiguity induced by degeneracy.
• Cumulative Path Metric: An additive metric penalizes deviations from the dominant decision, proportional to the log-likelihood gap. This favors paths that branch only when necessary and quantifies the global plausibility of the decoding trajectory.
• Efficient Scheduling: The learned Q-table (from RL-S) optimizes update order, exploiting syndrome-dependent local structure, with low-complexity updates at each expansion.
• Modularity: RL-LS integrates list exploration without altering the BP message update rules or the RL policy, preserving forward compatibility with advances in BP or RL scheduling.
• Practical Latency Management: The list size and the number of iterations are tunable, supporting hardware-efficient implementations and parallelization of list element processing.

Numerical Results

Experiments focus on depolarizing channel errors for prominent QLDPC codes: BB and A5 families. Key findings include:

• Error Correction Performance: RL-LS provides strong improvements in block error rate versus classical BP, guided decimation, and RL-S itself. For instance, RL-LS achieves block error rates competitive with RL-S but with an order-of-magnitude reduction in iteration budget, directly translating to lower decoding latency ((2606.20926), see Figure 1 and Table 1).
• Iteration Efficiency: RL-LS not only improves reliability but also reduces the average number of BP iterations required for convergence, outperforming all considered baselines at moderate and low depolarizing probabilities (see Tables 1–3).
• Scalability: Results for larger block lengths (BB288 and BB144) show RL-LS maintains its advantage, confirming its suitability for practical QLDPC deployments.
• Path Diversity and Convergence: The list-based strategy mitigates logical ambiguity and early trapping, accelerating convergence relative to both flooding and sequential BP approaches.

Theoretical and Practical Implications

The RL-LS decoder demonstrates the efficacy of combining learned sequential scheduling with list-based path exploration in QLDPC decoding. By explicitly branching at ambiguous decision points, RL-LS mitigates early deterministic misclassification, greatly reducing logical error probability. The cumulative metric provides an interpretable path-quality measure, supporting adaptive pruning and future integration with other BP enhancements.

Practically, RL-LS retains the low-complexity, localized message structure of BP, supporting hardware-friendly implementations, with tunable list size for balanced complexity-performance tradeoff. As the decoder architecture is modular, it can directly benefit from improvements in RL scheduling policies, path metric functions, or post-processing modules.

Theoretically, list exploration in BP decoding for quantum codes links the quantum decoding problem to classical sequential and list decoding paradigms (Fano, stack, list sequential decoding), drawing on well-understood principles in classical coding theory. The path metric formalizes how quantum degeneracy and ambiguous local beliefs impact decoding trajectories.

Future Directions

• Adaptive List Sizing: Online adjustment of list depth based on syndrome structure or residual mismatch statistics may further improve efficiency.
• Alternative Path Metrics: Incorporation of global evidence measures or syndrome-driven likelihoods could refine pruning and improve robustness.
• Integration with Neural BP and Trapping Set Mitigation: RL-LS is compatible with neural BP, trapping set-aware scheduling, and parallel bit-flipping continuations, offering synergy for further performance improvements.

Conclusion

The RL-LS BP decoder introduces controlled path diversity into learned sequential BP decoding for QLDPC codes, robustly addressing trapping sets and ambiguity from quantum degeneracy and short cycles. Empirical results show substantial gains in error-correction reliability and convergence speed versus conventional and learned baselines. The approach is modular, complexity-manageable, and theoretically well-grounded. RL-LS represents a significant advance in practical quantum BP decoding, and offers a foundation for further algorithmic improvements and hardware deployment in quantum error correction (2606.20926).