- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Almost linear decoder for optimal geometrically local quantum codes

Quinten Eggerickx1,2,3,*, Adam Wills4,5, Ting-Chun Lin6,5, Kristiaan De Greve1,2, and Min-Hsiu Hsieh5

  • *Contact author: quinten.eggerickx@imec.be
PDF

Phys. Rev. Research 7, 023300 – Published 24 June, 2025

DOI: https://doi.org/10.1103/sxdx-qbcz

Abstract

Geometrically local quantum codes, which are error-correction codes embedded in RD with checks acting only on qubits within a fixed spatial distance, have garnered significant interest. Recently, it has been demonstrated how to achieve geometrically local codes that maximize both the dimension and the distance, as well as the energy barrier of the code. In this work, we focus on the constructions involving subdivision, and we show that they have an almost linear time decoder, obtained by combining the decoder of the outer good qLDPC code and a generalized version of the Union-Find decoder. This provides the first decoder for an optimal geometrically local three-dimensional code. We demonstrate the existence of a finite threshold error rate under the code capacity noise model using a minimum weight perfect matching decoder. Furthermore, we argue that this threshold is also applicable to the decoder based on the generalized Union-Find algorithm.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (44)

  1. B. M. Terhal, Quantum error correction for quantum memories, Rev. Mod. Phys. 87, 307 (2015).
  2. L. Skoric, D. E. Browne, K. M. Barnes, N. I. Gillespie, and E. T. Campbell, Parallel window decoding enables scalable fault tolerant quantum computation, Nat. Commun. 14, 7040 (2023).
  3. D. Gottesman, Fault-tolerant quantum computation with constant overhead, arXiv:1310.2984.
  4. 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 (ACM Press, New York, NY, 2022), pp. 375–388.
  5. A. Leverrier and G. Zémor, Quantum tanner codes, in Proceedings of the 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, New York, 2022), pp. 872–883.
  6. I. Dinur, M.-H. Hsieh, T.-C. Lin, and T. Vidick, Good quantum ldpc codes with linear time decoders, in Proceedings of the 55th Annual ACM Symposium on Theory of Computing (ACM Press, New York, NY, 2023), pp. 905–918.
  7. S. Bravyi and B. Terhal, A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes, New J. Phys. 11, 043029 (2009).
  8. S. Bravyi, D. Poulin, and B. Terhal, Tradeoffs for reliable quantum information storage in 2D systems, Phys. Rev. Lett. 104, 050503 (2010).
  9. E. Portnoy, Local quantum codes from subdivided manifolds, arXiv:2303.06755.
  10. T.-C. Lin, A. Wills, and M.-H. Hsieh, Geometrically local quantum and classical codes from subdivision, arXiv:2309.16104.
  11. D. J. Williamson and N. Baspin, Layer codes, arXiv:2309.16503.
  12. X. Li, T.-C. Lin, and M.-H. Hsieh, Transform arbitrary good quantum ldpc codes into good geometrically local codes in any dimension, arXiv:2408.01769.
  13. N. Delfosse and N. H. Nickerson, Almost-linear time decoding algorithm for topological codes, Quantum 5, 595 (2021).
  14. B. J. Brown and D. J. Williamson, Parallelized quantum error correction with fracton topological codes, Phys. Rev. Res. 2, 013303 (2020).
  15. S. Kosen, H.-X. Li, M. Rommel, D. Shiri, C. Warren, L. Grönberg, J. Salonen, T. Abad, J. Biznárová, M. Caputo et al., Building blocks of a flip-chip integrated superconducting quantum processor, Quantum Sci. Technol. 7, 035018 (2022).
  16. D. Rosenberg, D. Kim, R. Das, D. Yost, S. Gustavsson, D. Hover, P. Krantz, A. Melville, L. Racz, G. O. Samach et al., 3D integrated superconducting qubits, npj Quantum Inf. 3, 42 (2017).
  17. D.-R. W. Yost, M. E. Schwartz, J. Mallek, D. Rosenberg, C. Stull, J. L. Yoder, G. Calusine, M. Cook, R. Das, A. L. Day et al., Solid-state qubits integrated with superconducting through-silicon vias, npj Quantum Inf. 6, 59 (2020).
  18. S. Zhang, Z. Li, H. Zhou, R. Li, S. Wang, K.-W. Paik, and P. He, Challenges and recent prospectives of 3d heterogeneous integration, e-Prime 2, 100052 (2022).
  19. Z. Cai, A. Siegel, and S. Benjamin, Looped pipelines enabling effective 3D qubit lattices in a strictly 2D device, PRX Quantum 4, 020345 (2023).
  20. F. van Riggelen-Doelman, C.-A. Wang, S. L. de Snoo, W. I. L. Lawrie, N. W. Hendrickx, M. Rimbach-Russ, A. Sammak, G. Scappucci, C. Déprez, and M. Veldhorst, Coherent spin qubit shuttling through germanium quantum dots, Nat. Commun. 15, 5716 (2024).
  21. M. De Smet, Y. Matsumoto, A.-M. J. Zwerver, L. Tryputen, S. L. de Snoo, S. V. Amitonov, A. Sammak, N. Samkharadze, Ö. Gül, R. N. M. Wasserman et al., High-fidelity single-spin shuttling in silicon, arXiv:2406.07267.
  22. M. B. Hastings, J. Haah, and R. O'Donnell, Fiber bundle codes: breaking the n 1/2 polylog (n) barrier for quantum ldpc codes, in Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (ACM Press, New York, NY, 2021), pp. 1276–1288.
  23. A. G. Fowler, Minimum weight perfect matching of fault-tolerant topological quantum error correction in average O(1) parallel time, Quantum Inf. Comput. 15, 145 (2015).
  24. Y. Wu and L. Zhong, Fusion blossom: Fast mwpm decoders for qec, in Proceedings of the 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE, New York, 2023), Vol. 1, pp. 928–938.
  25. 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.
  26. X. Tan, F. Zhang, R. Chao, Y. Shi, and J. Chen, Scalable surface-code decoders with parallelization in time, PRX Quantum 4, 040344 (2023).
  27. 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) (IEEE, New York, 2023), Vol. 1, pp. 916–927.
  28. H. Bombín, C. Dawson, Y.-H. Liu, N. Nickerson, F. Pastawski, and S. Roberts, Modular decoding: Parallelizable real-time decoding for quantum computers, arXiv:2303.04846.
  29. N. Delfosse and G. Zémor, Linear-time maximum likelihood decoding of surface codes over the quantum erasure channel, Phys. Rev. Res. 2, 033042 (2020).
  30. A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist, Phys. Rev. A 54, 1098 (1996).
  31. A. M. Steane, Error correcting codes in quantum theory, Phys. Rev. Lett. 77, 793 (1996).
  32. A. Yu. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. 303, 2 (2003).
  33. Kitaev surface code, in The Error Correction Zoo, edited by V. V. Albert and P. Faist (2025), https://errorcorrectionzoo.org/c/surface.
  34. N. Delfosse, P. Iyer, and D. Poulin, Generalized surface codes and packing of logical qubits, arXiv:1606.07116.
  35. T.-C. Lin and E. Portnoy (unpublished).
  36. S. Gu, C. A. Pattison, and E. Tang, An efficient decoder for a linear distance quantum ldpc code, in Proceedings of the 55th Annual ACM Symposium on Theory of Computing (ACM Press, New York, NY, 2023), pp. 919–932.
  37. A. Leverrier and G. Zémor, Efficient decoding up to a constant fraction of the code length for asymptotically good quantum codes, ACM Trans. Alg. (2024), doi:10.1145/3663763.
  38. A. Leverrier and G. Zémor, Efficient decoding up to a constant fraction of the code length for asymptotically good quantum codes, in Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms (SODA 2023), Florence, Italy (ACM, New York, 2023), Vol. 1, pp. 1216–1244.
  39. S. Gu, E. Tang, L. Caha, S. H. Choe, Z. He, and A. Kubica, Single-shot decoding of good quantum ldpc codes, Commun. Math. Phys. 405, 85 (2024).
  40. J. Hopcroft and R. Tarjan, Algorithm 447: Efficient algorithms for graph manipulation, Commun. ACM 16, 372 (1973).
  41. A. G. Fowler, Proof of finite surface code threshold for matching, Phys. Rev. Lett. 109, 180502 (2012).
  42. T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms, 3rd ed. (MIT Press, Cambridge, MA, 2009).
  43. B. A. Galler and M. J. Fisher, An improved equivalence algorithm, Commun. ACM 7, 301 (1964).
  44. R. E. Tarjan, Efficiency of a good but not linear set union algorithm, J. ACM 22, 215 (1975).

Outline

Information

Phys. Rev. Research 7, 023300– Published 24 June, 2025
  • Received 16 December 2024
  • Revised 15 April 2025
  • Accepted 11 May 2025
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation