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Zero transfer in continuous-time quantum walks

  • A. Sett
  • H. Pan
  • P. E. Falloon
  • J. B. WangEmail author
Article
  • 35 Downloads

Abstract

In this paper, we show how using complex-valued edge weights in a graph can completely suppress the flow of probability amplitude in a continuous-time quantum walk to specific vertices of the graph when the edge weights, graph topology, and initial state of the quantum walk satisfy certain conditions. The conditions presented in this paper are derived from the so-called chiral quantum walk, a variant of the continuous-time quantum walk which incorporates directional bias with respect to site transfer probabilities between vertices of a graph by using complex edge weights. We examine the necessity to break the time-reversal symmetry in order to achieve zero transfer in continuous-time quantum walks. We also consider the effect of decoherence on zero transfer and suggest that this phenomenon may be used to detect and quantify decoherence in the system.

Keywords

Chiral quantum walk Zero transfer Quantum stochastic walk Lindblad equation 

Notes

Acknowledgements

We would like to thank Aeysha Khalique for proof reading the manuscript and providing constructive suggestions.

References

  1. 1.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
  2. 2.
    Shor, P.W.: Algorithms for quantum computation: discrete logarithms and factoring. In: Foundations of Computer Science, 1994 Proceedings, 35th Annual Symposium on (IEEE), pp. 124–134 (1994)Google Scholar
  3. 3.
    Bennett, C.H., Brassard, G.: Quantum cryptography: public key distribution and coin tossing. Theor. Comput. Sci. 560, 7 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bennett, C.H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Phys. Rev. Lett. 70, 1895 (1993)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Lloyd, S.: Universal quantum simulators. Science 273, 1073–1078 (1996)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Feynman, R.P.: Simulating physics with computers. Int. J. Theor. Phys. 21, 467 (1982)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Childs, A.M.: Universal computation by quantum walk. Phys. Rev. Lett. 102, 180501 (2009)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Farhi, E., Gutmann, S.: Quantum computation and decision trees. Phys. Rev. A 58, 915 (1998)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Einstein, A.: Investigations on the Theory of the Brownian Movement. Courier Corporation, North Chelmsford (1956)zbMATHGoogle Scholar
  10. 10.
    Liu, X., Liu, J., Feng, Z.: Colorization using segmentation with random walk. In: International Conference on Computer Analysis of Images and Patterns (Springer), pp. 468–475 (2009)CrossRefGoogle Scholar
  11. 11.
    Ambainis, A.: Quantum walks and their algorithmic applications. Int. J. Quantum Inf. 1, 507 (2003)CrossRefGoogle Scholar
  12. 12.
    Wang, J.B., Manouchehri, K.: Physical Implementation of Quantum Walks. Springer, New York (2013)zbMATHGoogle Scholar
  13. 13.
    Izaac, J.A., Zhan, X., Bian, Z., Wang, K., Li, J., Wang, J.B., Xue, P.: Centrality measure based on continuous-time quantum walks and experimental realization. Phys. Rev. A 95, 032318 (2017)ADSCrossRefGoogle Scholar
  14. 14.
    Loke, T., Tang, J.W., Rodriguez, J., Small, M., Wang, J.B.: Comparing classical and quantum page ranks. Quantum Inf. Process. 16, 25 (2017)ADSCrossRefGoogle Scholar
  15. 15.
    Gamble, J.K., Friesen, M., Zhou, D., Joynt, R., Coppersmith, S.N.: Two-particle quantum walks applied to the graph isomorphism problem. Phys. Rev. A 81, 052313 (2010)ADSCrossRefGoogle Scholar
  16. 16.
    Xu, S., Sun, X., Wu, J., Zhang, W.-W., Arshed, N., Sanders, B.C.: Quantum walk on a chimera graph. New J. Phys. 20, 053039 (2018)ADSCrossRefGoogle Scholar
  17. 17.
    Childs, A.M., Goldstone, J.: Spatial search by quantum walk. Phys. Rev. A 70, 022314 (2004)ADSCrossRefGoogle Scholar
  18. 18.
    Li, Z.-J., Wang, J.B.: An analytical study of quantum walk through glued-tree graphs. J. Phys. A Math. Theor. 48, 355301 (2015)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Qiang, X., Loke, T., Montanaro, A., Aungskunsiri, K., Zhou, X., O’Brien, J.L., Wang, J.B., Matthews, J.C.: Efficient quantum walk on a quantum processor. Nat. Commun. 7, 11511 (2016)ADSCrossRefGoogle Scholar
  20. 20.
    Qiang, X., Zhou, X., Wang, J., Wilkes, C.M., Loke, T., O’Gara, S., Kling, L., Marshall, G.D., Santagati, R., Ralph, T.C., Wang, J.B., O’Brien, J.L., Thompson, M.G., Matthews, J.C.F.: Large-scale silicon quantum photonics implementing arbitrary two-qubit processing. Nat. Photonics (to appear, 2018)Google Scholar
  21. 21.
    Childs, A.M., Cleve, R., Deotto, E., Farhi, E., Gutmann, S., Spielman, D.A.: Exponential algorithmic speedup by a quantum walk. In: Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing (ACM), pp. 59–68 (2003)Google Scholar
  22. 22.
    Whitfield, J.D., Rodríguez-Rosario, C.A., Aspuru-Guzik, A.: Quantum stochastic walks: a generalization of classical random walks and quantum walks. Phys. Rev. A 81, 022323 (2010)ADSCrossRefGoogle Scholar
  23. 23.
    Sánchez-Burillo, E., Duch, J., Gómez-Gardenes, J., Zueco, D.: Quantum navigation and ranking in complex networks. Sci. Rep. 2, 605 (2012)ADSCrossRefGoogle Scholar
  24. 24.
    Zimborás, Z., Faccin, M., Kadar, Z., Whitfield, J.D., Lanyon, B.P., Biamonte, J.: Quantum transport enhancement by time-reversal symmetry breaking. Sci. Rep. 3, 2361 (2013)ADSCrossRefGoogle Scholar
  25. 25.
    Lu, D., Biamonte, J.D., Li, J., Li, H., Johnson, T.H., Bergholm, V., Faccin, M., Zimborás, Z., Laflamme, R., Baugh, J., et al.: Chiral quantum walks. Phys. Rev. A 93, 042302 (2016)ADSCrossRefGoogle Scholar
  26. 26.
    Godsil, C., Royle, G.F.: Algebraic Graph Theory, vol. 207. Springer, New York (2013)zbMATHGoogle Scholar
  27. 27.
    Von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17, 395 (2007)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Aharonov, Y., Bohm, D.: Significance of electromagnetic potentials in the quantum theory. Phys. Rev. 115, 485 (1959)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Falloon, P.E., Rodriguez, J., Wang, J.B.: QSWalk: a Mathematica package for quantum stochastic walks on arbitrary graphs. Comput. Phys. Commun. 217, 162 (2017)ADSCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of PhysicsThe University of Western AustraliaPerthAustralia
  2. 2.Department of PhysicsUniversity of MarylandCollege ParkUSA

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