Skip to main content
Springer Nature Link
Log in

Quantum chaos and random matrix theory for fidelity decay in quantum computations with static imperfections

  • Published:
The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics Aims and scope Submit manuscript

Abstract.

We determine the universal law for fidelity decay in quantum computations of complex dynamics in presence of internal static imperfections in a quantum computer. Our approach is based on random matrix theory applied to quantum computations in presence of imperfections. The theoretical predictions are tested and confirmed in extensive numerical simulations of a quantum algorithm for quantum chaos in the dynamical tent map with up to 18 qubits. The theory developed determines the time scales for reliable quantum computations in absence of the quantum error correction codes. These time scales are related to the Heisenberg time, the Thouless time, and the decay time given by Fermi’s golden rule which are well-known in the context of mesoscopic systems. The comparison is presented for static imperfection effects and random errors in quantum gates. A new convenient method for the quantum computation of the coarse-grained Wigner function is also proposed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

References

  1. A. Ekert, R. Josza, Rev. Mod. Phys. 68, 733 (1996)

    MathSciNet  Google Scholar 

  2. A. Steane, Rep. Prog. Phys. 61, 117 (1998)

    Google Scholar 

  3. M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge Univ. Press, Cambridge, 2000)

  4. D.P. Di Vincenzo, Science 270, 255 (1995)

    MathSciNet  Google Scholar 

  5. P.W. Shor, in Proc. 35th Annual Symposium on Foundation of Computer Science, edited by S. Goldwasser (IEEE Computer Society, Los Alamitos, CA, 1994), p. 124

  6. L.K. Grover, Phys. Rev. Lett. 79, 325 (1997)

    Article  Google Scholar 

  7. S. Lloyd, Science 273, 1073 (1996)

    MathSciNet  Google Scholar 

  8. G. Ortiz, J.E. Gubernatis, E. Knill, R. Laflamme, Phys. Rev. A 64, 22319 (2001)

    Google Scholar 

  9. B.V. Chirikov, F.M. Izrailev, D.L. Shepelyansky, Sov. Scient. Rev. C 2, 209 (1981); Physica D 33, 77 (1988)

    MATH  Google Scholar 

  10. F.M. Izrailev, Phys. Rep. 196, 299 (1990)

    Article  Google Scholar 

  11. R. Schack, Phys. Rev. A 57, 1634 (1998)

    MathSciNet  Google Scholar 

  12. B. Georgeot, D.L. Shepelyansky, Phys. Rev. Lett. 86, 2890 (2001)

    Article  Google Scholar 

  13. G. Benenti, G. Casati, S. Montangero, D.L. Shepelyansky, Phys. Rev. Lett. 87, 227901 (2001)

    Article  Google Scholar 

  14. A.D. Chepelianskii, D.L. Shepelyansky, Phys. Rev. A 66, 054301 (2002)

    Article  Google Scholar 

  15. A.A. Pomeransky, D.L. Shepelyansky, Phys. Rev. A (to appear), arXiv:quant-ph/0306203

  16. B. Georgeot, D.L. Shepelyansky, Phys. Rev. Lett. 86, 5393 (2001); Phys. Rev. Lett. 88, 219802 (2002)

    Article  Google Scholar 

  17. M. Terraneo, B. Georgeot, D.L. Shepelyansky, Eur. Phys. J. D 22, 127 (2003)

    Article  MathSciNet  Google Scholar 

  18. W.H. Zurek, Rev. Mod. Phys. 75, 715 (2003)

    Article  Google Scholar 

  19. B. Georgeot, D.L. Shepelyansky, Phys. Rev. E 62, 3504 (2000); Phys. Rev. E 62, 6366 (2000)

    Article  Google Scholar 

  20. V.V. Flambaum, Aust. J. Phys. 53, 489 (2000)

    MATH  Google Scholar 

  21. G.P. Berman, F. Borgonovi, F.M. Izrailev, V.I. Tsifrinovich, Phys. Rev. E 64, 056226 (2001)

    Article  Google Scholar 

  22. G. Benenti, G. Casati, D.L. Shepelyansky, Eur. Phys. J. D 17, 265 (2001)

    Article  Google Scholar 

  23. D. Braun, Phys. Rev. A 65, 042317 (2002)

    Article  Google Scholar 

  24. A. Peres, Phys. Rev. A 30, 1610 (1984)

    Article  MathSciNet  Google Scholar 

  25. D.L. Shepelyansky, Physica D 8, 208 (1983)

    Article  Google Scholar 

  26. G. Casati, B.V. Chirikov, I. Guarneri, D.L. Shepelyansky, Phys. Rev. Lett. 56, 2437 (1986)

    Article  Google Scholar 

  27. R.A. Jalabert, H.M. Pastawski, Phys. Rev. Lett. 86, 2490 (2001)

    Article  Google Scholar 

  28. P. Jacquod, P.G. Silvestrov, C.W.J. Beenakker, Phys. Rev. E 64, 055203 (2001)

    Article  Google Scholar 

  29. G. Veble, T. Prosen, Phys. Rev. Lett. 92, 034101 (2004)

    Article  Google Scholar 

  30. G. Benenti, G. Casati, Phys. Rev. E 66, 066205 (2002)

    Article  Google Scholar 

  31. T. Prosen, M. Znidaric, J. Phys. A 34, L681 (2001); J. Phys. A 35, 1455 (2002); T. Prosen, T.H. Seligman, M. Znidaric, Prog. Theor. Phys. Supp. 150, 200 (2003)

    Google Scholar 

  32. T. Kottos, D. Cohen, Europhys. Lett. 61, 431 (2003)

    Article  Google Scholar 

  33. N.R. Cerruti, S. Tomsovic, Phys. Rev. Lett. 88, 054103 (2002); J. Phys. A 36, 3451 (2003)

    Article  Google Scholar 

  34. Y. Adamov, I.V. Gornyi, A.D. Mirlin, Phys. Rev. E 67, 056217 (2003)

    Article  Google Scholar 

  35. C. Miguel, J.P. Paz, W.H. Zurek, Phys. Rev. Lett. 78, 3971 (1997)

    MATH  Google Scholar 

  36. M. Terraneo, D.L. Shepelyansky, Phys. Rev. Lett. 90, 257902 (2003)

    Article  Google Scholar 

  37. S. Bettelli, arXiv:quant-ph/0310152

  38. F.J. Dyson, J. Math. Phys. 3, 140 (1962)

    MATH  Google Scholar 

  39. M.L. Mehta, Random Matrices (Academic, New York, 1991)

  40. T. Guhr, A. Mueller-Groeling, H.A. Weidenmueller, Phys. Rep. 299, 189 (1998)

    Article  Google Scholar 

  41. E. Wigner, Phys. Rev. 40, 749 (1932)

    MATH  Google Scholar 

  42. S.-J. Chang, K.-J. Shi, Phys. Rev. A 34, 7 (1986)

    Article  MATH  Google Scholar 

  43. L.M.K. Vandersypen, M. Steffen, G. Breyta, C.S. Yannoni, M.H. Sherwood, I.L. Chuang, Nature 414, 883 (2001)

    Article  Google Scholar 

  44. Y.S. Weinstein, S. Lloyd, J. Emerson, D.G. Cory, Phys. Rev. Lett. 89, 157902 (2002)

    Article  Google Scholar 

  45. J. Emerson, Y.S. Weinstein, S. Lloyd, D.G. Cory, Phys. Rev. Lett. 89, 284102 (2002)

    Article  Google Scholar 

  46. B.V. Chirikov, Phys. Rep. 52, 263 (1979)

    Article  Google Scholar 

  47. S. Bullett, Com. Math. Phys. 107, 241 (1986)

    MathSciNet  MATH  Google Scholar 

  48. V.V. Vecheslavov, Zh. Eksp. Teor. Fiz. 119, 853 (2001), arXiv:nlin.CD/0005048; V.V. Vecheslavov, B.V. Chirikov, Zh. Eksp. Teor. Fiz. 120, 740 (2001), arXiv:nlin.CD/0202017

    Google Scholar 

  49. B.A. Muzykantskii, D.E. Khmelnitskii, Phys. Rev. B 51, 5481 (1995)

    Article  Google Scholar 

  50. A.D. Mirlin, Phys. Rep. 326, 259 (2000)

    Article  MathSciNet  Google Scholar 

  51. G. Casati, G. Maspero, D.L. Shepelyansky, Phys. Rev. E 56, R6233 (1997)

  52. D.V. Savin, V.V. Sokolov, Phys. Rev. E 56, R4911 (1997)

  53. K.M. Frahm, Phys. Rev. E 56, R6237 (1997)

  54. D. Gottesman, Phys. Rev. A 57, 127 (1998)

    Article  Google Scholar 

  55. D. Aharonov, M. Ben-Or, arXiv:quant-ph/9906129

  56. A. Steane, arXiv:quant-ph/0207119

  57. G. Benenti, G. Casati, S. Montangero, D.L. Shepelyansky, Eur. Phys. J. D 20, 293 (2002)

    Article  Google Scholar 

  58. B.L. Altshuler, B.I. Shklovskii, Zh. Eksp. Teor. Fiz. 91, 220 (1986) [Sov. Phys. JETP 64, 127 (1986)]

    Google Scholar 

  59. T. Gorin, T. Prosen, T. H. Seligman, preprint arXiv:nlin.CD/0311022v1

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Physique Théorique, UMR 5152 du CNRS, Université Paul Sabatier, 31062, Toulouse Cedex 4, France

    K. M. Frahm, R. Fleckinger & D. L. Shepelyansky

Authors
  1. K. M. Frahm
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. R. Fleckinger
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. D. L. Shepelyansky
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to D. L. Shepelyansky.

Additional information

Received: 13 December 2003, Published online: 18 March 2004

PACS:

03.67.Lx Quantum computation - 05.45.Pq Numerical simulations of chaotic systems - 05.45.Mt Quantum chaos; semiclassical methods

Rights and permissions

Reprints and permissions

About this article

Cite this article

Frahm, K.M., Fleckinger, R. & Shepelyansky, D.L. Quantum chaos and random matrix theory for fidelity decay in quantum computations with static imperfections. Eur. Phys. J. D 29, 139–155 (2004). https://doi.org/10.1140/epjd/e2004-00038-x

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjd/e2004-00038-x

Keywords