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  • Open Access

Beam Search Decoder for Quantum Low-Density Parity-Check Codes

Min Ye, Dave Wecker, and Nicolas Delfosse

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PRX Quantum 7, 033002 – Published 1 July, 2026

DOI: https://doi.org/10.1103/6k5x-ztqt

Abstract

We propose a decoder for quantum low-density parity-check (LDPC) codes based on a beam search heuristic guided by belief propagation (BP). Our beam search decoder applies to all quantum LDPC codes and achieves different speed-accuracy tradeoffs by tuning its parameters such as the beam width. We perform numerical simulations under circuit level noise for the [[144,12,12]] bivariate bicycle (BB) code at noise rate p=10−3 to estimate the logical error rate and the 99.9 percentile runtime and we compare with the BP-OSD decoder which has been the default quantum LDPC decoder for the past six years. A variant of our beam search decoder with a beam width of 64 achieves a 17× reduction in logical error rate. With a beam width of 8, we reach the same logical error rate as BP-OSD with a 26.2× reduction in the 99.9 percentile runtime. We identify the beam search decoder with beam width of 32 as a promising candidate for trapped ion architectures because it achieves a 5.6× reduction in logical error rate with a 99.9 percentile runtime per syndrome extraction round below 1 ms at p=5×10−4. Remarkably, this is achieved in software on a single core, without any parallelization or specialized hardware (FPGA, ASIC), suggesting one might only need three 32-core CPUs to decode a trapped ion quantum computer with 1000 logical qubits.

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Popular Summary

It is a widely held belief that fault-tolerant quantum computing will require massive, specialized supercomputing hardware to decode errors in real time. Our work challenges this assumption for architectures like trapped ions and neutral atoms. We demonstrate that decoding can be performed efficiently in software using standard, off-the-shelf hardware; specifically, a trapped ion computer with 1000 logical qubits could be decoded using only three commodity 32-core CPUs. We achieved this by designing a “beam search decoder” for quantum low-density parity-check codes. This proposed approach overcomes the limitations of the BP-OSD decoder, the default standard for the past six years, by simultaneously achieving higher accuracy and meeting the worst-case sub-millisecond latency requirements necessary for practical quantum error correction.

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References (64)

  1. R. Gallager, Low-density parity-check codes, IRE Trans. Inf. Theory 8, 21 (1962).
  2. D. J. MacKay, Information Theory, Inference and Learning Algorithms (Cambridge University Press, Cambridge, England, 2003).
  3. T. Richardson and R. Urbanke, Modern Coding Theory (Cambridge University Press, Cambridge, England, 2008).
  4. J. Pearl, Reverend bayes on inference engines: A distributed hierarchical approach, in Proceedings of the Second AAAI Conference on Artificial Intelligence, AAAI’82 (AAAI Press, 1982), pp. 133–136.
  5. D. J. C. MacKay, G. Mitchison, and P. L. McFadden, Sparse-graph codes for quantum error correction, IEEE Trans. Inf. Theory 50, 2315 (2004).
  6. J.-P. Tillich and G. Zémor, Quantum LDPC codes with positive rate and minimum distance proportional to the square root of the blocklength, IEEE Trans. Inf. Theory 60, 1193 (2013).
  7. A. A. Kovalev and L. P. Pryadko, Quantum Kronecker sum-product low-density parity-check codes with finite rate, Phys. Rev. A 88, 012311 (2013).
  8. N. P. Breuckmann and J. N. Eberhardt, Balanced product quantum codes, IEEE Trans. Inf. Theory 67, 6653 (2021).
  9. 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.
  10. A. Leverrier and G. Zémor, Quantum tanner codes, in Proceedings of the 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, 2022), pp. 872–883.
  11. O. Higgott and N. P. Breuckmann, Constructions and performance of hyperbolic and semi-hyperbolic floquet codes, PRX Quantum 5, 040327 (2024).
  12. 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 (London) 627, 778 (2024).
  13. M. Ye and N. Delfosse, Quantum error correction for long chains of trapped ions, Quantum 9, 1920 (2025).
  14. T. R. Scruby, T. Hillmann, and J. Roffe, High-threshold, low-overhead and single-shot decodable fault-tolerant quantum memory, PRX Quantum 7, 020310 (2026).
  15. M. A. Tremblay, N. Delfosse, and M. E. Beverland, Constant-overhead quantum error correction with thin planar connectivity, Phys. Rev. Lett. 129, 050504 (2022).
  16. A. Aydin, N. Delfosse, and E. Tham, Cyclic hypergraph product code, arXiv:2511.09683.
  17. D. Gottesman, Stabilizer codes and quantum error correction, Ph.D. thesis, California Institute of Technology, 1997.
  18. R. Tanner, A recursive approach to low complexity codes, IEEE Trans. Inf. Theory 27, 533 (1981).
  19. X.-Y. Hu, E. Eleftheriou, and D.-M. Arnold, Progressive edge-growth tanner graphs, in GLOBECOM’01, in Proceedings of the IEEE Global Telecommunications Conference (Cat. No. 01CH37270) (IEEE, New York, 2001), Vol. 2, pp. 995–1001.
  20. P. Panteleev and G. Kalachev, Degenerate quantum LDPC codes with good finite length performance, Quantum 5, 585 (2021).
  21. J. Roffe, D. R. White, S. Burton, and E. Campbell, Decoding across the quantum low-density parity-check code landscape, Phys. Rev. Res. 2, 043423 (2020).
  22. J. Roffe, LDPC: Python tools for low density parity check codes (2022).
  23. N. Delfosse, V. Londe, and M. E. Beverland, Toward a union-find decoder for quantum LDPC codes, IEEE Trans. Inf. Theory 68, 3187 (2022).
  24. S. Wolanski and B. Barber, Introducing ambiguity clustering: An accurate and efficient decoder for QLDPC codes, in Proceedings of the 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE, New York, 2024), Vol. 2, pp. 402–403.
  25. T. Hillmann, L. Berent, A. O. Quintavalle, J. Eisert, R. Wille, and J. Roffe, Localized statistics decoding for quantum low-density parity-check codes, Nat. Commun. 16, 8214 (2025).
  26. B. M. Terhal, Quantum error correction for quantum memories, Rev. Mod. Phys. 87, 307 (2015).
  27. A. Khalid, A. Silva, G. A. Dagnew, T. Dvir, O. Wertheim, M. Gruda, X. Kong, M. Kramer, Z. Webb, A. Scherer et al., Impacts of decoder latency on utility-scale quantum computer architectures, arXiv:2511.10633.
  28. D. Poulin and Y. Chung, On the iterative decoding of sparse quantum codes, Quantum Inf. Comput. 8, 987 (2008).
  29. J. Du Crest, M. Mhalla, and V. Savin, Stabilizer inactivation for message-passing decoding of quantum LDPC codes, in Proceedings of the 2022 IEEE Information Theory Workshop (ITW) (IEEE, New York, 2022), pp. 488–493.
  30. 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 Proceedings of the 2024 IEEE International Symposium on Information Theory (ISIT) (IEEE, New York, 2024), pp. 2478–2483.
  31. A. Gong, S. Cammerer, and J. M. Renes, Toward low-latency iterative decoding of QLDPC codes under circuit-level noise, arXiv:2403.18901.
  32. K. R. Ott, B. Hetényi, and M. E. Beverland, Decision-tree decoders for general quantum LDPC codes, arXiv:2502.16408.
  33. A. d. iOlius, I. E. Martinez, J. Roffe, and J. E. Martinez, An almost-linear time decoding algorithm for quantum LDPC codes under circuit-level noise, arXiv:2409.01440.
  34. T. Müller, T. Alexander, M. E. Beverland, M. Bühler, B. R. Johnson, T. Maurer, and D. Vandeth, Improved belief propagation is sufficient for real-time decoding of quantum memory, arXiv:2506.01779.
  35. T. Maurer, M. Bühler, M. Kröner, F. Haverkamp, T. Müller, D. Vandeth, and B. R. Johnson, Real-time decoding of the gross code memory with FPGAs, arXiv:2510.21600.
  36. S. Koutsioumpas, H. Sayginel, M. Webster, and D. E. Browne, Automorphism ensemble decoding of quantum LDPC codes, arXiv:2503.01738.
  37. M. Wang, A. Li, and F. Mueller, Fully parallelized bp decoding for quantum LDPC codes can outperform bp-osd, in Proceedings of the 2026 IEEE International Symposium on High Performance Computer Architecture (HPCA) (IEEE, New York, 2026), pp. 1–14.
  38. Y. Wu, B. Li, K. Chang, S. Puri, and L. Zhong, Minimum-weight parity factor decoder for quantum error correction, arXiv:2508.04969.
  39. L. A. Beni, O. Higgott, and N. Shutty, Tesseract: A search-based decoder for quantum error correction, arXiv:2503.10988.
  40. L. Valentini, D. Forlivesi, A. Talarico, and M. Chiani, Restart belief: A general quantum LDPC decoder, IEEE Commun. Lett. 30, 1185 (2026).
  41. A. Hughes, R. Srinivas, C. Löschnauer, H. Knaack, R. Matt, C. Ballance, M. Malinowski, T. Harty, and R. Sutherland, Trapped-ion two-qubit gates with >99.99% fidelity without ground-state cooling, arXiv:2510.17286.
  42. Source code of beam search decoder (2025). https://github.com/ionq-publications/BeamSearchDecoder.
  43. P. Das, C. A. Pattison, S. Manne, D. M. Carmean, K. M. Svore, M. Qureshi, and N. Delfosse, Afs: Accurate, fast, and scalable error-decoding for fault-tolerant quantum computers, in Proceedings of the 2022 IEEE International Symposium on High-Performance Computer Architecture (HPCA) (IEEE, New York, 2022), pp. 259–273.
  44. N. Liyanage, Y. Wu, A. Deters, and L. Zhong, Scalable quantum error correction for surface codes using FPGA, in Proceedings of the 2023 IEEE International Conference on Quantum Computing and Engineering (QCE), Vol. IEEE, New York, 2023), Vol. 1, pp. 916–927.
  45. N. Liyanage, Y. Wu, S. Tagare, and L. Zhong, FPGA-based distributed union-find decoder for surface codes, IEEE Trans. Quantum Eng. 5, 1 (2024).
  46. L. Caune, L. Skoric, N. S. Blunt, A. Ruban, J. McDaniel, J. A. Valery, A. D. Patterson, A. V. Gramolin, J. Majaniemi, K. M. Barnes et al., Demonstrating real-time and low-latency quantum error correction with superconducting qubits, arXiv:2410.05202.
  47. B. Barber, K. M. Barnes, T. Bialas, O. Buğdayc𝚤, E. T. Campbell, N. I. Gillespie, K. Johar, R. Rajan, A. W. Richardson, L. Skoric et al., A real-time, scalable, fast and resource-efficient decoder for a quantum computer, Nat. Electron. 8, 84 (2025).
  48. A. B. Ziad, A. Zalawadiya, C. Topal, J. Camps, G. P. Gehér, M. P. Stafford, and M. L. Turner, Local clustering decoder as a fast and adaptive hardware decoder for the surface code, Nat. Commun. 16, 11048 (2025).
  49. Y. Wu, N. Liyanage, and L. Zhong, Micro blossom: Accelerated minimum-weight perfect matching decoding for quantum error correction, in Proceedings of the 30th ACM International Conference on Architectural Support for Programming Languages and Operating Systems (2025), Vol. 2, pp. 639–654.
  50. F. Valentino, B. Branchini, D. Conficconi, D. Sciuto, and M. D. Santambrogio, QUEKUF: An FPGA union find decoder for quantum error correction on the toric code, ACM Trans. Reconfigurable Technol. Syst. 18, 1 (2025).
  51. S. Maurya, T. Maurer, M. Bühler, D. Vandeth, and M. E. Beverland, FPGA-tailored algorithms for real-time decoding of quantum LDPC codes, arXiv:2511.21660.
  52. H. Nishimori, Statistical Physics of Spin Glasses and Information Processing: An Introduction (Clarendon Press, 2001), Vol. 111.
  53. M. Mezard and A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009).
  54. E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, J. Math. Phys. (N.Y.) 43, 4452 (2002).
  55. N. Delfosse and G. Zémor, Quantum erasure-correcting codes and percolation on regular tilings of the hyperbolic plane, in 2010 IEEE Information Theory Workshop (IEEE, New York, 2010), pp. 1–5.
  56. H. Bombin, R. S. Andrist, M. Ohzeki, H. G. Katzgraber, and M. A. Martin-Delgado, Strong resilience of topological codes to depolarization, Phys. Rev. X 2, 021004 (2012).
  57. A. A. Kovalev and L. P. Pryadko, Spin glass reflection of the decoding transition for quantum error correcting codes, Quantum Inf. Comput. 15, 825 (2015).
  58. A. Kubica, M. E. Beverland, F. Brandão, J. Preskill, and K. M. Svore, Three-dimensional color code thresholds via statistical-mechanical mapping, Phys. Rev. Lett. 120, 180501 (2018).
  59. Y. Li and M. P. A. Fisher, Statistical mechanics of quantum error correcting codes, Phys. Rev. B 103, 104306 (2021).
  60. D. Vodola, M. Rispler, S. Kim, and M. Müller, Fundamental thresholds of realistic quantum error correction circuits from classical spin models, Quantum 6, 618 (2022).
  61. B. Placke, T. Rakovszky, N. P. Breuckmann, and V. Khemani, Topological quantum spin glass order and its realization in QLDPC codes, arXiv:2412.13248.
  62. B. Placke, G. M. Sommers, N. P. Breuckmann, T. Rakovszky, and V. Khemani, Expansion creates spin-glass order in finite-connectivity models: A rigorous and intuitive approach from the theory of LDPC codes, arXiv:2507.13342.
  63. L. H. English, D. J. Williamson, and S. D. Bartlett, Thresholds for postselected quantum error correction from statistical mechanics, Phys. Rev. Lett. 135, 120603 (2025).
  64. C. Gidney, Stim: A fast stabilizer circuit simulator, Quantum 5, 497 (2021).

Outline

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PRX Quantum 7, 033002– Published 1 July, 2026
  • Received 23 December 2025
  • Revised 12 May 2026
  • Accepted 26 May 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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