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LEGO_HQEC: A Software Tool for Analyzing Holographic Quantum Codes

Published 30 Oct 2024 in quant-ph | (2410.22861v1)

Abstract: Quantum error correction (QEC) is a crucial prerequisite for future large-scale quantum computation. Finding and analyzing new QEC codes, along with efficient decoding and fault-tolerance protocols, is central to this effort. Holographic codes are a recent class of QEC subsystem codes derived from holographic bulk/boundary dualities. In addition to exploring the physics of such dualities, these codes possess useful QEC properties such as tunable encoding rates, distance scaling competitive with topological codes, and excellent recovery thresholds. To allow for a comprehensive analysis of holographic code constructions, we introduce LEGO_HQEC, a software package utilizing the quantum LEGO formalism. This package constructs holographic codes on regular hyperbolic tilings and generates their stabilizer generators and logical operators for a specified number of seed codes and layers. Three decoders are included: an erasure decoder based on Gaussian elimination; an integer-optimization decoder; and a tensor-network decoder. With these tools, LEGO_HQEC thus enables future systematic studies regarding the utility of holographic codes for practical quantum computing.

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References (65)
  1. E. T. Campbell, B. M. Terhal, and C. Vuillot, Roads towards fault-tolerant universal quantum computation, Nature 549, 172 (2017).
  2. D. Gottesman, Fault-tolerant quantum computation with constant overhead, arXiv preprint arXiv:1310.2984  (2013).
  3. D. Gottesman, Opportunities and challenges in fault-tolerant quantum computation, arXiv preprint arXiv:2210.15844  (2022).
  4. D. Gottesman, Theory of fault-tolerant quantum computation, Phys. Rev. A 57, 127 (1998).
  5. J. Preskill, Reliable quantum computers, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 385 (1998).
  6. J. Preskill, Quantum computing in the nisq era and beyond, Quantum 2, 79 (2018).
  7. D. A. Lidar and T. A. Brun, Quantum error correction (Cambridge university press, 2013).
  8. M. M. Wilde, Quantum information theory (Cambridge university press, 2013).
  9. M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge university press, 2010).
  10. D. Gottesman, Stabilizer codes and quantum error correction (California Institute of Technology, 1997).
  11. V. V. Albert, Bosonic coding: introduction and use cases, arXiv preprint arXiv:2211.05714  (2022).
  12. N. P. Breuckmann and J. N. Eberhardt, Quantum low-density parity-check codes, PRX Quantum 2, 040101 (2021).
  13. A. Jahn and J. Eisert, Holographic tensor network models and quantum error correction: a topical review, Quantum Science and Technology 6, 033002 (2021).
  14. S. Huang and K. R. Brown, Between shor and steane: A unifying construction for measuring error syndromes, Phys. Rev. Lett. 127, 090505 (2021).
  15. C. Chamberland and M. E. Beverland, Flag fault-tolerant error correction with arbitrary distance codes, Quantum 2, 53 (2018).
  16. N. Delfosse, B. W. Reichardt, and K. M. Svore, Beyond single-shot fault-tolerant quantum error correction, IEEE Transactions on Information Theory 68, 287 (2021).
  17. H. Bombín, Single-shot fault-tolerant quantum error correction, Physical Review X 5, 031043 (2015a).
  18. D. P. DiVincenzo and P. W. Shor, Fault-tolerant error correction with efficient quantum codes, Physical review letters 77, 3260 (1996).
  19. D. P. DiVincenzo and P. Aliferis, Effective fault-tolerant quantum computation with slow measurements, Physical review letters 98, 020501 (2007).
  20. R. Chao and B. W. Reichardt, Quantum error correction with only two extra qubits, Physical review letters 121, 050502 (2018).
  21. P. Aliferis, D. Gottesman, and J. Preskill, Quantum accuracy threshold for concatenated distance-3 codes, arXiv preprint quant-ph/0504218  (2005).
  22. E. Knill, Quantum computing with realistically noisy devices, Nature 434, 39 (2005).
  23. P. W. Shor, Fault-tolerant quantum computation, in Proceedings of 37th conference on foundations of computer science (IEEE, 1996) pp. 56–65.
  24. N. Delfosse and B. W. Reichardt, Short shor-style syndrome sequences, arXiv preprint arXiv:2008.05051  (2020).
  25. H. Bombín, Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes, New Journal of Physics 17, 083002 (2015b).
  26. J. T. Anderson, G. Duclos-Cianci, and D. Poulin, Fault-tolerant conversion between the steane and reed-muller quantum codes, Phys. Rev. Lett. 113, 080501 (2014).
  27. A. Kubica and M. E. Beverland, Universal transversal gates with color codes: A simplified approach, Phys. Rev. A 91, 032330 (2015).
  28. T. Jochym-O’Connor and R. Laflamme, Using concatenated quantum codes for universal fault-tolerant quantum gates, Physical review letters 112, 010505 (2014).
  29. C. Chamberland, T. Jochym-O’Connor, and R. Laflamme, Thresholds for universal concatenated quantum codes, Physical review letters 117, 010501 (2016).
  30. B. M. Terhal, Quantum error correction for quantum memories, Reviews of Modern Physics 87, 307 (2015).
  31. A. Almheiri, X. Dong, and D. Harlow, Bulk locality and quantum error correction in ads/cft, Journal of High Energy Physics 2015, 1 (2015).
  32. J. Maldacena, The large-n limit of superconformal field theories and supergravity, International journal of theoretical physics 38, 1113 (1999).
  33. E. Witten, Anti de sitter space and holography, arXiv preprint hep-th/9802150  (1998).
  34. L. Boyle, M. Dickens, and F. Flicker, Conformal quasicrystals and holography, Phys. Rev. X 10, 011009 (2020).
  35. A. Jahn, Z. Zimborás, and J. Eisert, Tensor network models of ads/qcft, Quantum 6, 643 (2022).
  36. A. Jahn, Z. Zimborás, and J. Eisert, Central charges of aperiodic holographic tensor-network models, Phys. Rev. A 102, 042407 (2020).
  37. C. Piveteau, C. T. Chubb, and J. M. Renes, Tensor-network decoding beyond 2d, PRX Quantum 5, 040303 (2024).
  38. C. T. Chubb and S. T. Flammia, Statistical mechanical models for quantum codes with correlated noise, Annales de l’Institut Henri Poincaré D 8, 269 (2021).
  39. J. Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics/Springer-Verlag 149 (1994).
  40. A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist, Phys. Rev. A 54, 1098 (1996).
  41. A. M. Steane, Simple quantum error-correcting codes, Phys. Rev. A 54, 4741 (1996).
  42. C. Cao and B. Lackey, Quantum lego: Building quantum error correction codes from tensor networks, PRX Quantum 3, 020332 (2022).
  43. T. Farrelly, D. K. Tuckett, and T. M. Stace, Local tensor-network codes, New Journal of Physics 24, 043015 (2022b).
  44. M. Enríquez, I. Wintrowicz, and K. Życzkowski, Maximally entangled multipartite states: a brief survey, in Journal of Physics: Conference Series, Vol. 698 (IOP Publishing, 2016) p. 012003.
  45. M. Steinberg, S. Feld, and A. Jahn, Holographic codes from hyperinvariant tensor networks, Nature Communications 14, 10.1038/s41467-023-42743-z (2023).
  46. S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge theory correlators from non-critical string theory, Physics Letters B 428, 105 (1998).
  47. C. H. Bennett, D. P. DiVincenzo, and J. A. Smolin, Capacities of quantum erasure channels, Phys. Rev. Lett. 78, 3217 (1997).
  48. R. Penrose, A generalized inverse for matrices, in Mathematical proceedings of the Cambridge philosophical society, Vol. 51 (Cambridge University Press, 1955) pp. 406–413.
  49. E. H. Moore, On the reciprocal of the general algebraic matrix, Bulletin of the american mathematical society 26, 294 (1920).
  50. S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits, Phys. Rev. A 70, 052328 (2004).
  51. F. Le Gall, Powers of tensors and fast matrix multiplication, in Proceedings of the 39th international symposium on symbolic and algebraic computation (2014) pp. 296–303.
  52. S. Bravyi, M. Suchara, and A. Vargo, Efficient algorithms for maximum likelihood decoding in the surface code, Phys. Rev. A 90, 032326 (2014).
  53. L. Berent, L. Burgholzer, and R. Wille, Software tools for decoding quantum low-density parity-check codes, in Proceedings of the 28th Asia and South Pacific Design Automation Conference (2023) pp. 709–714.
  54. C. Gidney, Stim: a fast stabilizer circuit simulator, Quantum 5, 497 (2021).
  55. D. K. Tuckett, Tailoring surface codes: Improvements in quantum error correction with biased noise, Ph.D. thesis, University of Sydney (2020), (qecsim: https://github.com/qecsim/qecsim).
  56. Qiskit Community, Qiskit: An open-source framework for quantum computing (2017).
  57. B. Eastin and E. Knill, Restrictions on transversal encoded quantum gate sets, Physical review letters 102, 110502 (2009).
  58. A. Jahn, M. Steinberg, and J. Eisert, Holographic codes with many logical qubits, In Preparation  (2024).
  59. R. Shen, Y. Wang, and C. Cao, Quantum lego and xp stabilizer codes, arXiv preprint arXiv:2310.19538  (2023).
  60. M. A. Webster, B. J. Brown, and S. D. Bartlett, The xp stabiliser formalism: a generalisation of the pauli stabiliser formalism with arbitrary phases, Quantum 6, 815 (2022).
  61. E. Kubischta and I. Teixeira, Family of quantum codes with exotic transversal gates, Phys. Rev. Lett. 131, 240601 (2023).
  62. V. P. Roychowdhury and F. Vatan, On the structure of additive quantum codes and the existence of nonadditive codes, arXiv preprint quant-ph/9710031  (1997).
  63. M. Grassl and T. Beth, A note on non-additive quantum codes, arXiv preprint quant-ph/9703016  (1997).
  64. G. Evenbly, Hyperinvariant tensor networks and holography, Phys. Rev. Lett. 119, 141602 (2017).
  65. M. Steinberg and J. Prior, Conformal properties of hyperinvariant tensor networks, Scientific Reports 12, 532 (2022).

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