Quantum Information Processing

, Volume 12, Issue 1, pp 303–320 | Cite as

Mixing-time and large-decoherence in continuous-time quantum walks on one-dimension regular networks

  • R. Radgohar
  • S. Salimi


In this paper, we study mixing and large decoherence in continuous-time quantum walks on one dimensional regular networks, which are constructed by connecting each node to its 2l nearest neighbors (l on either side). In our investigation, the nodes of network are represented by a set of identical tunnel-coupled quantum dots in which decoherence is induced by continuous monitoring of each quantum dot with nearby point contact detector. To formulate the decoherent CTQWs, we use Gurvitz model and then calculate probability distribution and the bounds of instantaneous and average mixing times. We show that the mixing times are linearly proportional to the decoherence rate. Moreover, adding links to cycle network, in appearance of large decoherence, decreases the mixing times.


Quantum walk Decoherence Mixing-time One-dimension regular networks 


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  1. 1.
    Feynman R.P., Leighton R.B., Sands M.: Feynman Lectures on Physics. Addison Wesley, Boston (1964)Google Scholar
  2. 2.
    Farhi E., Gutmann S.: Quantum computation and decision trees. Phys. Rev. A 58, 915–928 (1998)MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J.: In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing (STOC’01), pp. 37–49. ACM Press, New York (2001)Google Scholar
  4. 4.
    Konno N.: One-dimensional discrete-time quantum walks on random environments. Quant. Inf. Process. 8(5), 387–399 (2009)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Krovi H., Brun T.A.: Quantum walks on quotient graphs. Phys. Rev. A 75, 062332 (2007)MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    Mülken O., Blumen A.: Continuous-time quantum walks in phase space. Phys. Rev. A 73, 012105 (2006)MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    Chandrashekar, C.M.: Discrete time quantum walk model for single and entangled particles to retain entanglement in coin space. arXiv: quant-ph/0609113V4 (2006)Google Scholar
  8. 8.
    Gottlieb A.D.: Convergence of continuous-time quantum walks on the line. Phys. Rev. E 72, 047102 (2005)ADSCrossRefGoogle Scholar
  9. 9.
    Avraham D., Bollt E., Tamon C.: One-dimensional continuous-time quantum walks. Quant. Inf. Process. 3, 295 (2004)MATHCrossRefGoogle Scholar
  10. 10.
    Salimi S.: Continuous-time quantum walks on star graphs. Ann. Phys. 324, 1185–1193 (2009)MathSciNetADSMATHCrossRefGoogle Scholar
  11. 11.
    Xu X.: Exact analytical results for quantum walks on star graph. J. Phys. A. Math. Theor. 42, 115205 (2009)ADSCrossRefGoogle Scholar
  12. 12.
    Salimi S., Jafarizadeh M.: Continuous-time classical and quantum random walk on direct product of Cayley graphs. Commun. Theor. Phys. 51, 1003–1009 (2009)MathSciNetADSMATHCrossRefGoogle Scholar
  13. 13.
    Salimi S.: Continuous-time quantum walks on semi-regular spidernet graphs via quantum probability theory. Quant. Inf. Process. 9, 75–91 (2008)MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    Salimi S.: Quantum central limit theorem for continuous-time quantum walks on odd graphs in quantum probability theory. Int. J. Theor. Phys. 47, 3298–3309 (2008)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Jafarizadeh M.A., Salimi S.: Investigation of continuous-time quantum walk via spectral distribution associated with adjacency matrix. Ann. Phys. 322, 1005–1033 (2007)MathSciNetADSMATHCrossRefGoogle Scholar
  16. 16.
    Konno N.: Continuous-time quantum walks on trees in quantum probability theory. Infin. Dimens. Anal. Quant. Probab. Relat. Top. 9(2), 287–297 (2006)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Xu X.: Coherent exciton transport and trapping on long-range interacting cycles. Phys. Rev. E 79, 011117 (2009)ADSCrossRefGoogle Scholar
  18. 18.
    Konno N.: Continuous-time quantum walks on ultrametric spaces. Int. J. Quant. Inf. 4(6), 1023–1035 (2006)MATHCrossRefGoogle Scholar
  19. 19.
    Drezgić M., Hines A.P., Sarovar M., Sastry Sh.: Complete Characterization of mixing time for the continuous quantum walk on the hypercube with Markovian decoherence model. Quant. Inf. Comput 9, 854 (2009)Google Scholar
  20. 20.
    Kendon V.: Decoherence in quantum walks—a review. Math. Struct. Comput. Sci. 17(6), 1169–1220 (2006)MathSciNetGoogle Scholar
  21. 21.
    Strauch F.W.: Reexamination of decoherence in quantum walks on the hypercube. Phys. Rev. A 79, 032319 (2009)ADSCrossRefGoogle Scholar
  22. 22.
    Alagic G., Russell A.: Decoherence in quantum walks on the hypercube. Phys. Rev. A 72, 062304 (2005)ADSCrossRefGoogle Scholar
  23. 23.
    Romanelli A., Siri R., Abal G., Auyuanet A., Donangelo R.: Decoherence in the quantum walk on the line. J. Phys. A 347, 137–152 (2005)MathSciNetGoogle Scholar
  24. 24.
    Kendon V., Tregenna B.: Decoherence can be useful in quantum walks. Phy. Rev. A 67, 042315 (2003)ADSCrossRefGoogle Scholar
  25. 25.
    Salimi S., Radgohar R.: Mixing and decoherence in continuous-time quantum walks on long-range interacting cycles. J. Phys. A Math. Theor. 42, 475302 (2009)MathSciNetADSCrossRefGoogle Scholar
  26. 26.
    Salimi S., Radgohar R.: The effect of large decoherence on mixing time in continuous-time quantum walks on long-range interacting cycles. J. Phys. B. At. Mol. Opt. Phys. 43, 025503 (2010)ADSCrossRefGoogle Scholar
  27. 27.
    Fedichkin L., Solenov D., Tamon C.: Mixing and decoherence in continuous-time quantum walks on cycles. Quant. Inf. Comput. 6(3), 263–276 (2006)MathSciNetMATHGoogle Scholar
  28. 28.
    Dür W.: Quantum walks in optical lattices. Phys. Rev. A 66, 052319 (2002)ADSCrossRefGoogle Scholar
  29. 29.
    Côté R.: Quantum random walk with Rydberg atoms in an optical lattice. New J. Phys. 8, 156 (2006)ADSCrossRefGoogle Scholar
  30. 30.
    Salimi S., Radgohar R.: The effect of decoherence on mixing time in continuous-time quantum walks on one-dimensional regular networks. Int. J. Quant. Inf. 8(5), 795–806 (2010)MATHCrossRefGoogle Scholar
  31. 31.
    Xu X.: Continuous-time quantum walks on one-dimensional regular networks. Phys. Rev. E 77, 061127 (2008)ADSCrossRefGoogle Scholar
  32. 32.
    Strogatz S.H., Stewart I.: Coupled oscillators and biological synchronization. Sci. Am. 269, 102 (1993)CrossRefGoogle Scholar
  33. 33.
    Wiesenfeld K.: New results on frequency-locking dynamics of disordered Josephson arrays. Phys. B 222, 315 (1996)ADSCrossRefGoogle Scholar
  34. 34.
    Belykh I.V., Belykh V.N., Hasler M.: Connection graph stability method for synchronized coupled chaotic systems. Phys. D 195, 159–187 (2004)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Watts D.J., Strogatz S.H.: Collective dynamics of ’small-world’ networks. Nature 393, 440–442 (1998)ADSCrossRefGoogle Scholar
  36. 36.
    Childs A.M., Goldstone J.: Spatial search by quantum walk. Phys. Rev. A 70, 022314–022324 (2004)MathSciNetADSCrossRefGoogle Scholar
  37. 37.
    Mülken O., Blumen A.: Slow transport by continuous time quantum walks. Phys. Rev. E 71, 016101–016106 (2005)ADSCrossRefGoogle Scholar
  38. 38.
    Volta A., Mülken O., Blumen A.: Quantum transport on two-dimensional regular graphs. J. Phys. A 39, 14997–15012 (2006)MathSciNetADSMATHCrossRefGoogle Scholar
  39. 39.
    Montroll E.W., Weiss G.H.: Random walks on lattices. II. J. Math. Phys 6, 167–181 (1965)MathSciNetADSCrossRefGoogle Scholar
  40. 40.
    Childs A.M., Farhi E., Gutmann S.: An example of the difference between quantum and classical random walks. Quant. Inf. Process. 1, 35 (2002)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Mülken O., Blumen A.: Spacetime structures of continuous-time quantum walks. Phys. Rev. E 71, 036128 (2005)ADSCrossRefGoogle Scholar
  42. 42.
    Ziman J.M.: Principles of the Theory of Solids. Cambridge University Press, Cambridge (1972)Google Scholar
  43. 43.
    Solenov D., Fedichkin L.: Continuous-time quantum walks on a cycle graph. Phys. Rev. A 73, 012313 (2006)MathSciNetADSCrossRefGoogle Scholar
  44. 44.
    de la Torre A.C., Mártin H.O., Goyeneche D.: Quantum diffusion on a cyclic one-dimensional lattice. Phys. Rev. E 68, 031103 (2003)ADSCrossRefGoogle Scholar
  45. 45.
    Pioro-Ladriere M., Abolfath R., Zawadzki P., Lapointe J., Studenikin S.A., Sachrajda A.S., Hawrylak P.: Charge sensing of an artificial H 2+ molecule in lateral quantum dots. Phys. Rev. B 72, 125307 (2005)ADSCrossRefGoogle Scholar
  46. 46.
    Hines A.P., Stamp P.C.E.: Quantum walks, quantum gates, and quantum computers. Phys. Rev. A 75, 062321 (2007)MathSciNetADSCrossRefGoogle Scholar
  47. 47.
    Gurvitz S.A.: Rate equations for quantum transport in multidot systems. Phys. Rev. B 57, 6602 (1998)ADSCrossRefGoogle Scholar
  48. 48.
    Gurvitz S.A.: Measurements with a noninvasive detector and dephasing mechanism. Phys. Rev. B 56, 15215 (1997)ADSCrossRefGoogle Scholar
  49. 49.
    Abramowitz M., Stegun I.A.: Handbook of Mathematical Functions. Dover, New York (1972)MATHGoogle Scholar
  50. 50.
    Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: Quantum walks on graphs. In: Proceedings of ACM Symposium on Theory of Computation (STOC 01), pp. 50–59 (2001)Google Scholar
  51. 51.
    Arfken G.B., Weber H.J.: Mathematical Methods for Physicists, Chapter 5. Harcourt Academic Press, San Diego (1972)Google Scholar
  52. 52.
    Gurvitz S.A.: Quantum description of classical apparatus: zeno effect and decoherence. Quant. Inf. Process. 2, 15 (2003)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Physics, Faculty of ScienceUniversity of KurdistanSanandajIran

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