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Implementation of one-qubit gates using optimal control theory for a dissipative system

  • Regular Article - Mesoscopic and Nanoscale Systems
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Abstract

The main objective of this paper is optimal control of decoherence and dissipation in a system of single qubit. First, we use the Bloch–Redfield formalism to derive the master equation describing the evolution of the qubit state parameterized by vectors in the Bloch sphere. The dissipative effect is described by the driven spin-boson model. By use of the optimal control methodology, we determine the fields that generate Not gate and Hadamard gate. Our optimal control has two components. To solve our optimal control problem, we use the techniques of automatic differentiation (AD) as an alternative to the Pontryagin’s minimum principle (PMP). The later is less straightforward since the master equation is complex. According to our numerical results, we conjecture that in the regime of weak system–bath coupling and low temperature, the driven spin-boson model is quasi-controllable.

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Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: The paper contents are purely theoretical, and did not need any data.]

References

  1. M.A. Nielson, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000)

    Google Scholar 

  2. C.H. Bennett, Quantum information and computation. Phys. Today 48, 24 (1995)

    Article  Google Scholar 

  3. N. Khaneja, R. Brockett, S.J. Glaser, Phys. Rev. A 63, 032308 (2001)

    Article  ADS  Google Scholar 

  4. N. Khaneja, S.J. Glaser, Chem. Phys. 267, 11–23 (2001)

    Article  Google Scholar 

  5. J. Zhang, J. Vala, S. Sastry, R.B. Whaley, Phys. Rev. A 67, 042313 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  6. A. Carlini, A. Hossoya, T. Koike, Y. Okudaira, Phys. Rev. Lett. 96, 060503 (2006)

    Article  ADS  Google Scholar 

  7. H. Jirari, Phys. Rev. A 102, 012613 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  8. M. Grifoni, P. Hänggi, Phys. Rep. 304, 229 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  9. U. Weiss, Quantum Dissipative Systems (World Scientific, Singapore, 1999)

    Book  MATH  Google Scholar 

  10. P. Zanardi, M. Rasetti, Phys. Rev. Lett. 79, 3306 (1997)

    Article  ADS  Google Scholar 

  11. D.A. Lidar, I.L. Chuang, K.B. Whaley, Phys. Rev. Lett. 81, 2594 (1999)

    Article  ADS  Google Scholar 

  12. J. Preskill, Proc. R. Soc. Lond. A 454, 385 (1998)

    Article  ADS  Google Scholar 

  13. E. Knill, R. Laflamme, L. Viola, Phys. Rev. Lett. 84, 2525 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  14. J. Wang, H. Wiseman, Phys. Rev. A 64, 063810 (2001)

    Article  ADS  Google Scholar 

  15. L. Viola, S. Lloyd, Phys. Rev. A 58, 2733 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  16. L. Viola, E. Knill, S. Lloyd, Phys. Rev. Lett. 82, 2417 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  17. L. Viola, E. Knill, S. Lloyd, Phys. Rev. Lett. 83, 4888 (1999)

    Article  ADS  Google Scholar 

  18. L. Viola, E. Knill, S. Lloyd, Phys. Rev. Lett. 85, 3520 (2000)

    Article  ADS  Google Scholar 

  19. P. Zanardi, Phys. Lett. A 258, 77 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  20. D. Vitali, P. Tombesi, Phys. Rev. A 59, 4178 (1999)

    Article  ADS  Google Scholar 

  21. C. Uchiyama, M. Aihara, Ibid 66, 032313 (2002)

    Google Scholar 

  22. M.S. Byrd, D.A. Lidar, Quantum Inf. Process. 1, 19 (2002)

    Article  MathSciNet  Google Scholar 

  23. M.S. Byrd, D.A. Lidar, Phys. Rev. A 67, 012324 (2003)

    Article  ADS  Google Scholar 

  24. V. Protopopescu, R. Perez, C. D’Helon, J. Schmulen, J. Phys. A Math. Gen. 36, 2175 (2003)

    Article  ADS  Google Scholar 

  25. M. Itano. Wayne, Phys. Rev. A 41, 2295 (1990)

    Article  Google Scholar 

  26. P. Facchi, S. Tasaki, S. Pascazio, H. Nakazato, A. Tokuse, D.A. Lidar, Phys. Rev. A 71, 022302 (2005)

    Article  ADS  Google Scholar 

  27. L. Viola, J. Mod. Opt. 51, 2357 (2004)

    Article  ADS  Google Scholar 

  28. X. Hu, W. Pötz, Phys. Rev. Lett. 82, 3116 (1999)

    Article  ADS  Google Scholar 

  29. L.S. Pontryagin, The Mathematical Theory of the Optimal Process (Wiley-Interscience, New York, 1962)

    Google Scholar 

  30. L. Hascoët, V. Pascual, A.C.M. Trans, Math. Softw. 39, 20 (2013)

    Article  Google Scholar 

  31. T. Tamayo-Mendoza, C. Kreisbek, R. Lindh, A. Aspuru-Gusik, ACS Cent. Sci. 4, 559 (2018)

    Article  Google Scholar 

  32. A.E. Bryson, Y.C. Ho, Applied Optimal Control (Hemisphere, New York, 1975)

    Google Scholar 

  33. Shi, H. Rabitz, J. Chem. Phys. 92, 364 (1988)

    Article  ADS  Google Scholar 

  34. M. Grace et al., J. Phys. B At. Mol. Opt. Phys. 40, S103–S125 (2007)

    Article  ADS  Google Scholar 

  35. G.D. Mahan, Many Particles Physics (Plenum Press, New York, 1999)

    Google Scholar 

  36. B. Krummheuer, V.M. Axt, T. Kuhn, Phys. Rev. B. 65, 195313 (2002)

    Article  ADS  Google Scholar 

  37. U. Hohenester, G. Stadler, Phys. Rev. Lett 92, 196801 (2004)

    Article  ADS  Google Scholar 

  38. J. Zhang, W. Pötz, Phys. Rev. B 48, 11583 (1993)

    Article  ADS  Google Scholar 

  39. I. Rousochatzakis, Y. Ajiro, H. Mitamura, P. Kögerler, M. Luban, Phys. Rev. Lett 94, 147204 (2005)

    Article  ADS  Google Scholar 

  40. A.E. Allahverdyan, R.S. Garcià, Th.M. Nieuwenhuizen, Phys. Rev. Lett. 93, 260404 (2005)

    Article  Google Scholar 

  41. Yu. Makhlin, G. Schön, S. Shnirman, Rev. Mod. Phys. 73, 357–400 (2001)

    Article  ADS  Google Scholar 

  42. P. Nalbach, Phys. Rev. A 90, 042112 (2014)

    Article  ADS  Google Scholar 

  43. L. Arceci et al., Phys. Rev. B 96, 054301 (2017)

    Article  ADS  Google Scholar 

  44. A.J. Leggett, S. Chakravarty, A.T. Dorsey, M.P.A. Fisher, A. Garg, W. Zwerger, Rev. Mod. Phys. 59, 1 (1987)

    Article  ADS  Google Scholar 

  45. G. Wendin, V.S. Shumeiko, in Handbook of Theoretical and Computational Nanothecnology, vol. 3, ed. by M. Rirth, W. Schommers (ASP , Los Angeles, 2006), pp. 223–309

  46. J.M. Martinis, S. Nam, J. Aumentado, C. Urbina, Phys. Rev. Lett. 89, 117901 (2002)

    Article  ADS  Google Scholar 

  47. M. Neeley et al., Nat. Phys. 4, 523 (2008)

    Article  Google Scholar 

  48. E. Hoskinson et al. Phys. Rev. Lett. 102, 097004 (2009)

  49. H. Jirari, F.W.J. Hekking, O. Buisson, Europhys. Lett. 87, 28004 (2009)

    Article  ADS  Google Scholar 

  50. J.F. Poyatos, J.J. Cirac, P. Zoller, Phys. Rev. Lett. 78, 390 (1997)

    Article  ADS  Google Scholar 

  51. M. Thorwart, P. Hänggi, Phys. Rev. A 65, 012309–1 (2001)

    Article  ADS  Google Scholar 

  52. H.P. Breuer, F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002)

    MATH  Google Scholar 

  53. M. Morillo, C. Denk, R.I. Cukier, Chem. Phys. 212, 157 (1996)

    Article  Google Scholar 

  54. K. Blum, Density Matrix Theory and Applications, 2nd edn. (Plenum Press, New York, 1996)

    Book  MATH  Google Scholar 

  55. L. Hartmann, I. Goychuk, M. Grifoni, P. Hänggi, Phys. Rev. E 61, R4687 (2000)

    Article  ADS  Google Scholar 

  56. G.H. Golub, C. Van Loan, Matrix Computations, 2nd edn. (John Hopkins University Press, Baltimore, 1989)

    MATH  Google Scholar 

  57. H. Jirari, Europhys. Lett 87, 40003 (2009)

    Article  ADS  Google Scholar 

  58. C. Zhu, R.H. Byrd, J. Nocedal, ACM Trans. Math. Softw. 23(4), 550–560 (1997)

  59. H. Geen, R. Freeman, Band selective radiofrequency pulses. J. Magn. Reson. 58, 442–457 (1984)

    ADS  Google Scholar 

  60. V. Ramakrishna, M.V. Salapaka, M. Dahleh, H. Rabitz, A. Pierce, Phys. Rev. A. 51, 960 (1995)

    Article  ADS  Google Scholar 

  61. S.G. Schirmer, H. Fu, A.I. Solomon, Phys. Rev. A. 63, 063410 (2001)

    Article  ADS  Google Scholar 

  62. D. D’Alessandro, preprint arXiv:quant-ph/0110120 (2001)

Download references

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    H. Jirari

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Jirari, H. Implementation of one-qubit gates using optimal control theory for a dissipative system. Eur. Phys. J. B 94, 217 (2021). https://doi.org/10.1140/epjb/s10051-021-00221-9

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