Quantum Information Processing

, Volume 9, Issue 1, pp 75–91 | Cite as

Continuous-time quantum walks on semi-regular spidernet graphs via quantum probability theory

Article
First Online:
Received:
Accepted:

Abstract

We analyze continuous-time quantum and classical random walk on spidernet lattices. In the framework of Stieltjes transform, we obtain density of states, which is an efficiency measure for the performance of classical and quantum mechanical transport processes on graphs, and calculate the spacetime transition probabilities between two vertices of the lattice. Then we analytically show that there are two power law decays ∼ t −3 and ∼ t −1.5 at the beginning of the transport for transition probability in the continuous-time quantum and classical random walk, respectively. This results illustrate the decay of quantum mechanical transport processes is quicker than that of the classical one. Due to the result, the characteristic time t c , which is the time when the first maximum of the probabilities occur on an infinite graph, for the quantum walk is shorter than that of the classical walk. Therefore, we can interpret that the quantum transport speed on spidernet is faster than that of the classical one. In the end, we investigate the results by numerical analysis for two examples.

Keywords

Continuous-time quantum walk Spidernet graphs Spectral distribution 

PACS

03.65.Ud 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aharonov Y., Davidovich L., Zagury N.: Quantum random walks. Phys. Rev. A 48, 1687–1690 (1993)CrossRefPubMedADSGoogle Scholar
  2. 2.
    Childs, A., Cleve, R., Deotto, E., Farhi, E., Gutmann, S., Spielman, D.: Exponential algorithmic speedup by quantum walk. Proceedings of 35th Annual Symposium on Theory of Computing, ACM Press, 59–68 (2003).Google Scholar
  3. 3.
    Shenvi N., Kempe J., Whaley K.B.: Quantum random-walk search algorithm. Phys. Rev. A 67, 052307–052318 (2003)CrossRefADSGoogle Scholar
  4. 4.
    Kempe J.: Quantum random walks: an introductory overview. Contemp. Phys. 44, 307–327 (2003)CrossRefADSGoogle Scholar
  5. 5.
    Feynman R., Leighton R., Sands M.: The Feynman Lectures on Physics, vol. 3. Addison-Wesley, London (1965)Google Scholar
  6. 6.
    Farhi E., Gutmann S.: Quantum computation and decision trees. Phys. Rev. A 58, 915–928 (1998)CrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Childs M., Farhi E., Gutmann S.: An example of the difference between quantum and classical random walks. Quantum. Inf. Process. 1, 35–43 (2002)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Ambainis, A., Bach, E., Nayak, A., Viswanath, A., Watrous, J.: One-dimensional quantum walks. Conference Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, 37–49 (2001)Google Scholar
  9. 9.
    Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: Quantum walks on graphs. Conference Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, 50–59 (2001)Google Scholar
  10. 10.
    Moore, C., Russell, A.: Quantum walks on the hypercube. Proceedings of the 6th International Workshop on Randomization and Approximation in Computer Science (RANDOM’02), 164 (2002).Google Scholar
  11. 11.
    Kempe, J.: Quantum Random Walks Hit Exponentially Faster. Proceedings of 7th International Workshop on Randomization and Approximation Techniques in Computer Science (RANDOM’03), 354–369 (2003).Google Scholar
  12. 12.
    Krovi H., Brun T.A.: Quantum walks on quotient graphs. Phys. Rev. A 75, 062332–062345 (2007)CrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Konno N.: Continuous-time quantum walks on ultrametric spaces. Int. J. Quantum Inf. 4, 1023–1035 (2006)CrossRefMATHGoogle Scholar
  14. 14.
    Jafarizadeh M.A., Salimi S.: Investigation of continuous-time quantum walk via spectral distribution associated with adjacency matrix. Ann. Phys. 322, 1005–1033 (2007)CrossRefMathSciNetADSMATHGoogle Scholar
  15. 15.
    Jafarizadeh M.A., Salimi S.: Investigation of continuous-time quantum walk via modules of Bose-Mesner and Terwilliger algebras. J. Phys. A 39, 13295–13323 (2006)CrossRefMathSciNetADSMATHGoogle Scholar
  16. 16.
    Jafarizadeh M.A., Sufiani R., Salimi S., Jafarizadeh S.: Investigation of continuous-time quantum walk by using Krylov subspace-Lanczos algorithm. Eur. Phys. J. B 59, 199–216 (2007)CrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Konno N.: Continuous-time quantum walks on trees in quantum probability theory. Inf. Dim. Anal. Quantum Probab. Rel. Topics 9, 287–297 (2006)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    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)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Salimi S.: Study of continuous-time quantum walks on quotient graphs via quantum probability theory. Int. J. Quantum Inf. 6, 945–957 (2008)CrossRefMATHGoogle Scholar
  20. 20.
    Vogtle, F. (ed.): Dendrimers. Springer, Berlin (1998)Google Scholar
  21. 21.
    Mulken O., Bierbaum V., Blumen A.: Coherent exciton transport in dendrimers and continuous-time quantum walks. J. Chem. Phys. 124, 124905 (2006)CrossRefPubMedADSGoogle Scholar
  22. 22.
    Mülken O., Blumen A.: Efficiency of quantum and classical transport on graphs. Phys. Rev. E 73, 066111–066117 (2006)CrossRefGoogle Scholar
  23. 23.
    Mülken, O.: Inefficient quantum walks on networks: the role of the density of states. arXiv:0710.3453 (2007)Google Scholar
  24. 24.
    Xu X.P.: Continuous-time quantum walks on one-dimensional regular networks. Phys. Rev. E 77, 061127–061135 (2008)CrossRefADSGoogle Scholar
  25. 25.
    Obata N.: Quantum probabilistic approach to spectral analysis of star graphs. Interdiscip. Inf. Sci. 10, 41–52 (2004)MathSciNetMATHGoogle Scholar
  26. 26.
    Hora A., Obata N.: Quantum Probability and Spectral Analysis of Graphs. Springer, Heidelberg (2007)MATHGoogle Scholar
  27. 27.
    Urakawa H.: The Cheeger constant, the heat kernel, and the Green kernel of an infinite graph. Monatsh. Math. 138, 225 (2003)CrossRefMathSciNetMATHGoogle Scholar
  28. 28.
    Weiss G.H.: Aspects and Applications of the Random Walk. Elsevier Science Pub Co, Amsterdam (1994)MATHGoogle Scholar
  29. 29.
    van Kampen N.: Stochastic Processes in Physics and Chemistry. 3rd edn. North Holland, Amsterdam (2007)Google Scholar
  30. 30.
    Bender C.M., Orszag S.A.: Advanced Mathematical Methods for Scientist and Engineers. McGraw-Hill, Inc., New York (1978)Google Scholar
  31. 31.
    Kesten H.: Symmetric random walks on groups. Trans. Am. Math. Soc. 92, 336 (1959)CrossRefMathSciNetMATHGoogle Scholar
  32. 32.
    Chihara T.S.: An Introduction to Orthogonal Polynomials. Gordon and Breach, London (1978)MATHGoogle Scholar
  33. 33.
    Shohat J.A., Tamarkin J.D.: The Problem of Moments, 4th edn. American Mathematical Society, Providence, RI (1943)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of KurdistanSanandajIran

Personalised recommendations