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Single-shot decoding of good quantum LDPC codes (2306.12470v2)

Published 21 Jun 2023 in quant-ph, cs.IT, and math.IT

Abstract: Quantum Tanner codes constitute a family of quantum low-density parity-check (LDPC) codes with good parameters, i.e., constant encoding rate and relative distance. In this article, we prove that quantum Tanner codes also facilitate single-shot quantum error correction (QEC) of adversarial noise, where one measurement round (consisting of constant-weight parity checks) suffices to perform reliable QEC even in the presence of measurement errors. We establish this result for both the sequential and parallel decoding algorithms introduced by Leverrier and Z\'emor. Furthermore, we show that in order to suppress errors over multiple repeated rounds of QEC, it suffices to run the parallel decoding algorithm for constant time in each round. Combined with good code parameters, the resulting constant-time overhead of QEC and robustness to (possibly time-correlated) adversarial noise make quantum Tanner codes alluring from the perspective of quantum fault-tolerant protocols.

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References (43)
  1. P. W. Shor, Scheme for reducing decoherence in quantum computer memory, Physical Review A 52, R2493 (1995).
  2. A. M. Steane, Error correcting codes in quantum theory, Physical Review Letters 77, 793 (1996a).
  3. D. Gottesman, Class of quantum error-correcting codes saturating the quantum Hamming bound, Physical Review A 54, 1862 (1996).
  4. A. Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics 303, 2 (2003).
  5. H. Bombin and M. A. Martin-Delgado, Topological quantum distillation, Phys. Rev. Lett. 97, 180501 (2006).
  6. H. Bombín and M. Martin-Delgado, Exact topological quantum order in D=3𝐷3D=3italic_D = 3 and beyond: Branyons and brane-net condensates, Physical Review B 75, 075103 (2007).
  7. A. Kubica, The ABCs of the Color Code: A Study of Topological Quantum Codes as Toy Models for Fault-Tolerant Quantum Computation and Quantum Phases Of Matter, Ph.D. thesis, Caltech (2018).
  8. S. Bravyi and B. Terhal, A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes, New Journal of Physics 11, 043029 (2009).
  9. S. Bravyi, D. Poulin, and B. Terhal, Tradeoffs for reliable quantum information storage in 2D systems, Phys. Rev. Lett. 104, 050503 (2010).
  10. N. Baspin and A. Krishna, Quantifying nonlocality: How outperforming local quantum codes is expensive, Physical Review Letters 129, 050505 (2022).
  11. N. P. Breuckmann and J. N. Eberhardt, Quantum low-density parity-check codes, PRX Quantum 2, 040101 (2021a).
  12. S. Evra, T. Kaufman, and G. Zémor, Decodable quantum LDPC codes beyond the n𝑛\sqrt{n}square-root start_ARG italic_n end_ARG distance barrier using high-dimensional expanders, SIAM Journal on Computing 0, FOCS20 (0).
  13. M. B. Hastings, J. Haah, and R. O’Donnell, Fiber bundle codes: breaking the n1/2⁢polylog⁢(n)superscript𝑛12polylog𝑛n^{1/2}\text{polylog}(n)italic_n start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT polylog ( italic_n ) barrier for quantum LDPC codes, in Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (2021) pp. 1276–1288.
  14. N. P. Breuckmann and J. N. Eberhardt, Balanced product quantum codes, IEEE Transactions on Information Theory 67, 6653 (2021b).
  15. P. Panteleev and G. Kalachev, Quantum LDPC codes with almost linear minimum distance, IEEE Transactions on Information Theory 68, 213 (2022a).
  16. 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.
  17. A. Leverrier and G. Zémor, Quantum Tanner codes, in 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, 2022) pp. 872–883.
  18. B. M. Terhal, Quantum error correction for quantum memories, Rev. Mod. Phys. 87, 307 (2015).
  19. 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 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) (SIAM, 2023) pp. 1216–1244.
  20. P. Shor, Fault-tolerant quantum computation, in Proceedings of 37th Conference on Foundations of Computer Science (IEEE Comput. Soc. Press, 1996) pp. 56–65.
  21. H. Bombín, Single-shot fault-tolerant quantum error correction, Phys. Rev. X 5, 031043 (2015).
  22. O. Fawzi, A. Grospellier, and A. Leverrier, Constant overhead quantum fault-tolerance with quantum expander codes, in 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, 2018).
  23. A. Kubica and M. Vasmer, Single-shot quantum error correction with the three-dimensional subsystem toric code, Nature Communications 13, 6272 (2022).
  24. J. C. Bridgeman, A. Kubica, and M. Vasmer, Lifting topological codes: Three-dimensional subsystem codes from two-dimensional anyon models (2023), arXiv:2305.06365.
  25. H. Bombín, Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes, New Journal of Physics 17, 083002 (2015).
  26. E. T. Campbell, A theory of single-shot error correction for adversarial noise, Quantum Science and Technology 4, 025006 (2019).
  27. Y. Fujiwara, Ability of stabilizer quantum error correction to protect itself from its own imperfection, Physical Review A 90, 062304 (2014).
  28. A. Ashikhmin, C. Y. Lai, and T. A. Brun, Quantum Data-Syndrome Codes, IEEE Journal on Selected Areas in Communications 38, 449 (2020).
  29. N. Delfosse, B. W. Reichardt, and K. M. Svore, Beyond single-shot fault-tolerant quantum error correction, IEEE Transactions on Information Theory 68, 287 (2022).
  30. A. Leverrier and G. Zémor, Decoding quantum tanner codes, IEEE Transactions on Information Theory , 1 (2023).
  31. E. Ben-Sasson and M. Sudan, Robust locally testable codes and products of codes, Random Structures & Algorithms 28, 387 (2006).
  32. G. Kalachev and P. Panteleev, Two-sided robustly testable codes (2022), arXiv:2206.09973.
  33. A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist, Phys. Rev. A 54, 1098 (1996).
  34. A. Steane, Multiple-particle interference and quantum error correction, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 452, 2551 (1996b).
  35. E. Knill, Quantum computing with realistically noisy devices, Nature 434, 39 (2005).
  36. D. Gottesman, Fault-tolerant quantum computation with constant overhead, Quantum Info. Comput. 14, 1338–1372 (2014).
  37. A. Grospellier, Constant time decoding of quantum expander codes and application to fault-tolerant quantum computation, Ph.D. thesis, Sorbonne Université (2019).
  38. D. Aharonov and L. Eldar, Quantum locally testable codes, SIAM Journal on Computing 44, 1230 (2015).
  39. M. B. Hastings, Quantum codes from high-dimensional manifolds, in 8th Innovations in Theoretical Computer Science Conference (ITCS 2017) (Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2017).
  40. A. Leverrier, V. Londe, and G. Zémor, Towards local testability for quantum coding, Quantum 6, 661 (2022).
  41. A. Wills, T.-C. Lin, and M.-H. Hsieh, General distance balancing for quantum locally testable codes (2023), arXiv:2305.00689.
  42. M. A. Tremblay, N. Delfosse, and M. E. Beverland, Constant-overhead quantum error correction with thin planar connectivity, Physical Review Letters 129, 050504 (2022).
  43. C. A. Pattison, A. Krishna, and J. Preskill, Hierarchical memories: Simulating quantum LDPC codes with local gates (2023), arXiv:2303.04798.
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