pith. sign in

arxiv: 2606.20926 · v1 · pith:B4JWJNSTnew · submitted 2026-06-18 · 💻 cs.IT · math.IT· quant-ph

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

Mohsen Moradi , Taejoon Kim , Remi A. Chou This is my paper

Pith reviewed 2026-06-26 15:03 UTC · model grok-4.3

classification 💻 cs.IT math.ITquant-ph
keywords quantum LDPC codesbelief propagation decodingreinforcement learninglist decodingsequential schedulingdepolarizing channelerror correction
0
0 comments X

The pith

A reinforcement learning list-sequential BP decoder improves decoding performance for quantum LDPC codes over the depolarizing channel.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a decoder for quantum LDPC codes that uses reinforcement learning to select the order of variable-node updates in belief propagation. It augments this sequential schedule with list search that also explores a competing branch at each step by softly biasing messages toward the second-most-likely Pauli symbol. The two trajectories are ranked and pruned with a cumulative path metric. The approach targets the convergence failures that standard BP exhibits on these codes because of short cycles and degeneracy. A sympathetic reader would care because more reliable decoders would make quantum error correction on these strong candidate codes more practical.

Core claim

The authors introduce the RL-LS BP decoder by extending the RL-S framework with list-based search. At each step a learned policy chooses the next variable node to update; the decoder retains the main trajectory while also creating a competing branch by softly biasing the post-update LLR pair toward the second-most-likely Pauli symbol, recomputing incident local BP messages, and assigning that symbol to the visited node. Candidate trajectories are ranked and pruned using a proposed cumulative path metric. Numerical results on representative QLDPC benchmark codes over the depolarizing channel show that the method improves the decoding performance of the underlying decoder and compares favorabl

What carries the argument

The RL-LS BP decoder, which pairs a learned policy for sequential variable-node scheduling with list exploration of a softly biased second-best symbol branch that is ranked by cumulative path metric.

If this is right

  • The decoder raises the success rate of belief-propagation decoding on the tested QLDPC codes over the depolarizing channel.
  • List exploration of a second-best symbol branch yields better trajectories than the single learned scheduling path alone.
  • The combined method outperforms or matches several existing BP-based decoders on the same benchmark instances.
  • Learned sequential scheduling plus list search together mitigate the convergence problems caused by short cycles and degeneracy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same list-augmented scheduling idea could be tested on non-QLDPC quantum codes that also exhibit poor BP convergence.
  • Combining the RL-LS trajectories with a post-processing step such as ordered-statistics decoding might produce still lower error rates.
  • The cumulative path metric itself could be studied in isolation to see how much of the gain comes from ranking rather than from the learned policy.
  • Scaling the approach to larger code lengths would test whether the learned policy continues to generalize when the Tanner graph grows.

Load-bearing premise

The learned policy for variable-node scheduling generalizes across codes and noise levels and the list exploration with soft biasing toward the second-most-likely symbol reliably produces better trajectories than the single learned path.

What would settle it

Executing the RL-LS decoder on the same representative QLDPC benchmark codes over the depolarizing channel and measuring a frame error rate or bit error rate no lower than that achieved by the base RL-S decoder or other BP methods would disprove the claimed performance gain.

Figures

Figures reproduced from arXiv: 2606.20926 by Mohsen Moradi, Remi A. Chou, Taejoon Kim.

Figure 1
Figure 1. Figure 1: Tanner graph for the 12-variable CSS example. The highlighted variable nodes illustrate one RL-LS decoding step: v3 and v11 have already been explored in the current trajectory, and v7 is the current variable node where the local list expansion is carried out. All other variable nodes keep the same local RL state in this expansion step. Finally, suppose that after the update the ordinary child has residual… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the block error rate performance of our proposed RL-LS decoder with other decoders for the [[288, 12, 18]] BB code over the depolarizing channel. often require a substantially smaller average number of iterations to converge, and even under the same cap Imax, they typically achieve significantly better error-correction performance than conventional flooding BP. Moreover, cluster-based variant… view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of the block error rate performance of our proposed RL-LS decoder with other decoders for the [[144, 12, 12]] BB code over the depolarizing channel. ing with list-based exploration can substantially improve error-correction performance while maintaining a favorable decoding-latency profile. Table II reports the corresponding average number of iterations for the decoding schemes shown in [PITH_F… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper proposes a reinforcement learning-based list sequential belief propagation (RL-LS BP) decoder for quantum LDPC codes. It extends the prior RL-S framework by incorporating list-based search: a learned policy selects the next variable node to update, the decoder retains the RL-S trajectory while exploring a competing branch via soft biasing of the post-update LLR pair toward the second-most likely Pauli symbol, and trajectories are ranked and pruned using a proposed cumulative path metric. Numerical results on representative QLDPC benchmark codes over the depolarizing channel are reported to show improved decoding performance relative to the underlying decoder and favorable comparisons with existing BP-based methods.

Significance. If the reported empirical gains are reproducible, the work offers a practical contribution to decoding QLDPC codes by combining learned variable-node scheduling with targeted list exploration, addressing convergence difficulties arising from short cycles and degeneracy. The explicit extension of the RL-S framework and the provision of numerical comparisons on standard benchmarks constitute strengths that allow readers to assess the incremental benefit.

minor comments (2)
  1. [Method] The description of the cumulative path metric and the soft-biasing rule for the second branch would benefit from an explicit equation or pseudocode block in the method section to clarify implementation details.
  2. [Numerical Results] The numerical results section should include a table or explicit listing of the exact code parameters (length, rate, girth), training hyperparameters, and number of Monte Carlo trials for each reported curve to support reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work on the RL-LS BP decoder and for recommending minor revision. The referee's description accurately reflects the extension of the RL-S framework with list-based search and the reported numerical results on benchmark QLDPC codes.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper proposes RL-LS as an explicit extension of the authors' prior RL-S framework and supports its performance claims solely through numerical simulations on representative QLDPC codes under depolarizing noise. No equations, fitted parameters, or uniqueness theorems are presented that reduce the claimed gains to quantities defined by the authors' own prior results; the central empirical improvement is independently verifiable on the stated benchmarks and does not rely on self-citation chains for its validity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the method rests on standard BP message-passing assumptions and RL policy learning whose details are not supplied.

pith-pipeline@v0.9.1-grok · 5733 in / 1208 out tokens · 27313 ms · 2026-06-26T15:03:57.900892+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

45 extracted references · 7 canonical work pages · 1 internal anchor

  1. [1]

    Scheme for reducing decoherence in quantum computer memory,

    P. W. Shor, “Scheme for reducing decoherence in quantum computer memory,” Physical Review A, vol. 52, no. 4, pp. R2493– R2496, 1995

    1995

  2. [2]

    Quantum error correction for quantum memo- ries,

    B. M. Terhal, “Quantum error correction for quantum memo- ries,” Reviews of Modern Physics, vol. 87, no. 2, pp. 307–346, 2015

    2015

  3. [3]

    Quantum low-density parity-check codes,

    N. P. Breuckmann and J. N. Eberhardt, “Quantum low-density parity-check codes,” PRX Quantum, vol. 2, no. 4, p. 040101, 2021

    2021

  4. [4]

    Quantum LDPC codes with positive rate and minimum distance proportional to the square root of the blocklength,

    J.-P. Tillich and G. Zémor, “Quantum LDPC codes with positive rate and minimum distance proportional to the square root of the blocklength,” IEEE Transactions on Information Theory, vol. 60, no. 2, pp. 1193–1202, 2014

    2014

  5. [5]

    Balanced product quantum codes,

    N. P. Breuckmann and J. N. Eberhardt, “Balanced product quantum codes,” IEEE Transactions on Information Theory, vol. 67, no. 10, pp. 6653–6674, 2021

    2021

  6. [6]

    Asymptotically good quantum and locally testable classical LDPC codes,

    P. Panteleev and G. Kalachev, “Asymptotically good quantum and locally testable classical LDPC codes,” in Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, 2022, pp. 375–388

    2022

  7. [7]

    Quantum Tanner codes,

    A. Leverrier and G. Zémor, “Quantum Tanner codes,” in 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), 2022, pp. 872–883

    2022

  8. [8]

    High-threshold and low-overhead fault-tolerant quantum memory,

    S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, “High-threshold and low-overhead fault-tolerant quantum memory,” Nature, vol. 627, pp. 778–782, 2024

    2024

  9. [9]

    Kasai, Breaking the Orthogonality Barrier in Quan- tum LDPC Codes (2026), arXiv:2601.08824 [quant-ph]

    K. Kasai, “Breaking the orthogonality barrier in quantum LDPC codes,” arXiv preprint arXiv:2601.08824, 2026

    work page arXiv 2026

  10. [10]

    Towards Ultra-High-Rate Quantum Error Correction with Reconfigurable Atom Arrays

    C. Zhao, C. Duckering, A. Gu, N. Maskara, and H. Zhou, “Towards ultra-high-rate quantum error correction with recon- figurable atom arrays,” arXiv preprint arXiv:2604.16209, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  11. [11]

    Decoding across the quantum LDPC code landscape,

    J. Roffe, D. R. White, S. Burton, and E. T. Campbell, “Decoding across the quantum LDPC code landscape,” Physical Review Research, vol. 2, no. 4, p. 043423, 2020

    2020

  12. [12]

    Low-density parity-check codes,

    R. Gallager, “Low-density parity-check codes,” IRE Transactions on Information Theory, vol. 8, no. 1, pp. 21–28, 1962

    1962

  13. [13]

    Near Shannon limit perfor- mance of low density parity check codes,

    D. J. C. MacKay and R. M. Neal, “Near Shannon limit perfor- mance of low density parity check codes,” Electronics Letters, vol. 32, no. 18, pp. 1645–1646, 1996

    1996

  14. [14]

    Factor graphs and the sum-product algorithm,

    F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Transactions on Information Theory, vol. 47, no. 2, pp. 498–519, 2001

    2001

  15. [15]

    The capacity of low- density parity-check codes under message-passing decoding,

    T. J. Richardson and R. L. Urbanke, “The capacity of low- density parity-check codes under message-passing decoding,” IEEE Transactions on Information Theory, vol. 47, no. 2, pp. 599–618, 2001

    2001

  16. [16]

    On the iterative decoding of sparse quantum codes,

    D. Poulin and Y. Chung, “On the iterative decoding of sparse quantum codes,” Quantum Information & Computation, vol. 8, no. 10, pp. 987–1000, 2008

    2008

  17. [17]

    Exploiting degeneracy in belief prop- agation decoding of quantum codes,

    K.-Y. Kuo and C.-Y. Lai, “Exploiting degeneracy in belief prop- agation decoding of quantum codes,” npj Quantum Information, vol. 8, no. 1, p. 111, 2022

    2022

  18. [18]

    Trapping sets of quantum LDPC codes,

    N. Raveendran and B. Vasić, “Trapping sets of quantum LDPC codes,” Quantum, vol. 5, p. 562, 2021

    2021

  19. [19]

    Stabilizer inactivation for message-passing decoding of quantum LDPC codes,

    J. du Crest, M. Mhalla, and V. Savin, “Stabilizer inactivation for message-passing decoding of quantum LDPC codes,” in 2022 IEEE Information Theory Workshop (ITW), 2022, pp. 488–493

    2022

  20. [20]

    Localized statistics decoding for quantum low- density parity-check codes,

    T. Hillmann, L. Berent, A. O. Quintavalle, J. Eisert, R. Wille, and J. Roffe, “Localized statistics decoding for quantum low- density parity-check codes,” Nature Communications, vol. 16, p. 8214, 2025

    2025

  21. [21]

    Neural belief propagation decoding of quantum LDPC codes using overcomplete check matrices,

    S. Miao, A. Schnerring, H. Li, and L. Schmalen, “Neural belief propagation decoding of quantum LDPC codes using overcomplete check matrices,” in 2023 IEEE Information Theory Workshop (ITW), 2023, pp. 215–220

    2023

  22. [22]

    Quaternary neural belief propagation decoding of quan- tum LDPC codes with overcomplete check matrices,

    ——, “Quaternary neural belief propagation decoding of quan- tum LDPC codes with overcomplete check matrices,” IEEE Access, vol. 13, pp. 25 637–25 649, 2025

    2025

  23. [23]

    Graph neural networks for enhanced decoding of quantum LDPC codes,

    A. Gong, S. Cammerer, and J. M. Renes, “Graph neural networks for enhanced decoding of quantum LDPC codes,” in 2024 IEEE International Symposium on Information Theory (ISIT), 2024, pp. 2700–2705

    2024

  24. [24]

    Enhanced message-passing decoding of degenerate quantum codes utilizing trapping set dynamics,

    D. Chytas, M. Pacenti, N. Raveendran, M. F. Flanagan, and B. Vasić, “Enhanced message-passing decoding of degenerate quantum codes utilizing trapping set dynamics,” IEEE Com- munications Letters, vol. 28, no. 3, pp. 444–448, 2024

    2024

  25. [25]

    Belief propagation decoding of quantum LDPC codes with guided decimation,

    H. Yao, W. A. Laban, C. Häger, A. G. i Amat, and H. D. Pfister, “Belief propagation decoding of quantum LDPC codes with guided decimation,” in 2024 IEEE International Symposium on Information Theory (ISIT), 2024, pp. 2478–2483

    2024

  26. [26]

    Layered decoding of quantum LDPC codes,

    J. du Crest, F. Garcia-Herrero, M. Mhalla, V. Savin, and J. Valls, “Layered decoding of quantum LDPC codes,” in 2023 12th International Symposium on Topics in Coding (ISTC), 2023, pp. 1–5

    2023

  27. [27]

    Sequential BP-based decoding of QLDPC codes,

    M. Moradi, S. Habib, V. Nourozi, and D. G. M. Mitchell, “Sequential BP-based decoding of QLDPC codes,” CoRR, vol. abs/2602.13420, 2026

    work page arXiv 2026

  28. [28]

    Informed dynamic scheduling for QLDPC codes,

    T.-H. Huang and Y.-L. Ueng, “Informed dynamic scheduling for QLDPC codes,” Quantum, vol. 10, p. 1967, 2026

    1967

  29. [29]

    Fully parallelized BP decoding for quantum LDPC codes can outperform BP-OSD,

    M. Wang, A. Li, and F. Mueller, “Fully parallelized BP decoding for quantum LDPC codes can outperform BP-OSD,” CoRR, vol. abs/2507.00254, 2025. 13

    work page arXiv 2025

  30. [30]

    List decoding and new bicycle code constructions for quantum LDPC codes,

    S. Rabeti and H. Mahdavifar, “List decoding and new bicycle code constructions for quantum LDPC codes,” CoRR, vol. abs/2511.02951, 2025

    work page arXiv 2025

  31. [31]

    Learned parallel bit- flipping sequential belief propagation decoding of quantum LDPC codes,

    M. Moradi, T. Kim, and R. A. Chou, “Learned parallel bit- flipping sequential belief propagation decoding of quantum LDPC codes,” 2026, manuscript

    2026

  32. [32]

    Replica shuffled iterative decoding,

    J. Zhang, Y. Wang, M. P. C. Fossorier, and J. S. Yedidia, “Replica shuffled iterative decoding,” in Proceedings of the 2005 IEEE International Symposium on Information Theory (ISIT), 2005, pp. 454–458

    2005

  33. [33]

    Learn- ing to decode quantum LDPC codes via belief propagation,

    M. Moradi, V. Nourozi, S. Habib, and D. G. M. Mitchell, “Learn- ing to decode quantum LDPC codes via belief propagation,” CoRR, vol. abs/2603.10192, 2026

    work page arXiv 2026

  34. [34]

    RELDEC: Reinforce- ment learning-based decoding of moderate length LDPC codes,

    S. Habib, A. Beemer, and J. Kliewer, “RELDEC: Reinforce- ment learning-based decoding of moderate length LDPC codes,” CoRR, vol. abs/2112.13934, 2021

    work page arXiv 2021

  35. [35]

    Enhancing belief propagation decoding of polar codes: A reinforcement learning approach,

    M. Moradi, S. Habib, and D. G. M. Mitchell, “Enhancing belief propagation decoding of polar codes: A reinforcement learning approach,” IEEE Communications Letters, vol. 29, no. 6, pp. 1285–1289, 2025

    2025

  36. [36]

    A heuristic discussion of probabilistic decoding,

    R. M. Fano, “A heuristic discussion of probabilistic decoding,” IEEE Transactions on Information Theory, vol. 9, no. 2, pp. 64– 74, Apr. 1963

    1963

  37. [37]

    Some sequential decoding procedures,

    K. S. Zigangirov, “Some sequential decoding procedures,” Prob- lems of Information Transmission, vol. 2, no. 4, pp. 1–10, 1966

    1966

  38. [38]

    Fast sequential decoding algorithm using a stack,

    F. Jelinek, “Fast sequential decoding algorithm using a stack,” IBM Journal of Research and Development, vol. 13, no. 6, pp. 675–685, Nov. 1969

    1969

  39. [39]

    List decoding of polar codes,

    I. Tal and A. Vardy, “List decoding of polar codes,” IEEE Transactions on Information Theory, vol. 61, no. 5, pp. 2213– 2226, May 2015

    2015

  40. [40]

    On sequential decoding metric function of polarization-adjusted convolutional (PAC) codes,

    M. Moradi, “On sequential decoding metric function of polarization-adjusted convolutional (PAC) codes,” IEEE Trans- actions on Communications, vol. 69, no. 12, pp. 7913–7922, Dec. 2021

    2021

  41. [41]

    Stabilizer codes and quantum error correction,

    D. Gottesman, “Stabilizer codes and quantum error correction,” Ph.D. dissertation, California Institute of Technology, 1997

    1997

  42. [42]

    Good quantum error- correcting codes exist,

    A. R. Calderbank and P. W. Shor, “Good quantum error- correcting codes exist,” Physical Review A, vol. 54, no. 2, pp. 1098–1105, 1996

    1996

  43. [43]

    Multiple-particle interference and quantum error correction,

    A. M. Steane, “Multiple-particle interference and quantum error correction,” Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, vol. 452, no. 1954, pp. 2551–2577, 1996

    1954

  44. [44]

    Good error-correcting codes based on very sparse matrices,

    D. J. C. MacKay, “Good error-correcting codes based on very sparse matrices,” IEEE Transactions on Information Theory, vol. 45, no. 2, pp. 399–431, 1999

    1999

  45. [45]

    Degenerate quantum LDPC codes with good finite length performance,

    P. Panteleev and G. Kalachev, “Degenerate quantum LDPC codes with good finite length performance,” Quantum, vol. 5, p. 585, 2021

    2021