Quantum Information Processing

, Volume 15, Issue 5, pp 1865–1896 | Cite as

Quantum walks on simplicial complexes

Article

Abstract

We construct a new type of quantum walks on simplicial complexes as a natural extension of the well-known Szegedy walk on graphs. One can numerically observe that our proposing quantum walks possess linear spreading and localization as in the case of the Grover walk on lattices. Moreover, our numerical simulation suggests that localization of our quantum walks reflects not only topological but also geometric structures. On the other hand, our proposing quantum walk contains an intrinsic problem concerning exhibition of non-trivial behavior, which is not seen in typical quantum walks such as Grover walks on graphs.

Keywords

Quantum walk Simplicial complexes Tethered and movable quantum walks 

References

  1. 1.
    Gudder, S.P.: Quantum Probability. Probability and Mathematical Statistics. Academic, Boston (1988)MATHGoogle Scholar
  2. 2.
    Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Path Integrals. Dover Publications, Inc., Mineola (2010). Emended edition, Emended and with a preface by Daniel F. StyerMATHGoogle Scholar
  3. 3.
    Konno, N.: Quantum random walks in one dimension. Quantum Inf. Process. 1(5), 345–354 (2002). 2003MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ambainis, A.: Quantum walks and their algorithmic applications. Int. J. Quantum Inf. 1(04), 507–518 (2003)CrossRefMATHGoogle Scholar
  5. 5.
    Konno, N.: Quantum walks. In: Quantum Potential Theory, Vol. 1954 of Lecture Notes in Math., pp. 309–452. Springer, Berlin (2008)Google Scholar
  6. 6.
    Ambainis, A.: Quantum walk algorithm for element distinctness. SIAM J. Comput. 37(1), 210–239 (2007). (electronic)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Ambainis, A., Kempe, J., Rivosh, A.: Coins make quantum walks faster. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1099–1108 (electronic). ACM, New York (2005)Google Scholar
  8. 8.
    Shenvi, N., Kempe, J., Whaley, K.B.: Quantum random-walk search algorithm. Phys. Rev. A 67(5), 052307 (2003)ADSCrossRefGoogle Scholar
  9. 9.
    Abal, G., Donangelo, R., Forets, M., Portugal, R.: Spatial quantum search in a triangular network. Math. Struct. Comput. Sci. 22(03), 521–531 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chandrashekar, C.M., Banerjee, S., Srikanth, R.: Relationship between quantum walks and relativistic quantum mechanics. Phys. Rev. A 81(6), 062340 (2010)ADSCrossRefGoogle Scholar
  11. 11.
    Strauch, F.W.: Relativistic effects and rigorous limits for discrete- and continuous-time quantum walks. J. Math. Phys. 48(8), 082102 (2007). 27ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Karski, M., Förster, L., Choi, J.-M., Steffen, A., Alt, W., Meschede, D., Widera, A.: Quantum walk in position space with single optically trapped atoms. Science 325(5937), 174–177 (2009)ADSCrossRefGoogle Scholar
  13. 13.
    Matsuoka, L., Yokoyama, K.: Physical implementation of quantum cellular automaton in a diatomic molecule. J. Comput. Theor. Nanosci. 10(7), 1617–1620 (2013)CrossRefGoogle Scholar
  14. 14.
    Zähringer, F., Kirchmair, G., Gerritsma, R., Solano, E., Blatt, R., Roos, C.F.: Realization of a quantum walk with one and two trapped ions. Phys. Rev. Lett. 104(10), 100503 (2010)CrossRefGoogle Scholar
  15. 15.
    Wang, J., Manouchehri, K.: Physical Implementation of Quantum Walks. Springer, Berlin (2013)MATHGoogle Scholar
  16. 16.
    Berry, S.D., Bourke, P., Wang, J.B.: qwviz: visualisation of quantum walks on graphs. Comput. Phys. Commun. 182(10), 2295–2302 (2011)ADSCrossRefGoogle Scholar
  17. 17.
    Watrous, J.: Quantum simulations of classical random walks and undirected graph connectivity. In: Proceedings of Fourteenth Annual IEEE Conference on Computational Complexity, 1999, pp. 180–187 (1999)Google Scholar
  18. 18.
    Higuchi, Yu., Konno, N., Sato, I., Segawa, E.: Quantum graph walks I: mapping to quantum walks. Yokohama Math. J. 59, 33–55 (2013)MathSciNetMATHGoogle Scholar
  19. 19.
    Szegedy, M.: Quantum speed-up of markov chain based algorithms. In: Proceedings of 45th Annual IEEE Symposium on Foundations of Computer Science, 2004, pp. 32–41 (2004)Google Scholar
  20. 20.
    Paparo, G.D., Martin-Delgado, M.A.: Google in a quantum network. Sci. Rep. 2, 444 (2012)Google Scholar
  21. 21.
    Paparo, G.D., Müller, M., Comellas, F., Martin-Delgado, M.A.: Quantum google in a complex network. Sci. Rep. 3, 2773 (2013)Google Scholar
  22. 22.
    Higuchi, Yu., Konno, N., Sato, I., Segawa, E.: Spectral and asymptotic properties of Grover walks on crystal lattices. J. Funct. Anal. 267(11), 4197–4235 (2014)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2010)CrossRefMATHGoogle Scholar
  24. 24.
    Meyer, D.A.: From quantum cellular automata to quantum lattice gases. J. Stat. Phys. 85(5–6), 551–574 (1996)ADSMathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Tregenna, B., Flanagan, W., Maile, R., Kendon, V.: Controlling discrete quantum walks: coins and initial states. New J. Phys. 5(1), 83 (2003)ADSCrossRefGoogle Scholar
  26. 26.
    Milnor, J.W., Stasheff, J.D.: Characteristic Classes. Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. Ann. Math. Stud. No. 76Google Scholar
  27. 27.
    Matsue, K., Ogurisu, O., Segawa, E.: Quantum walks on cubical sets : construction and asymptotic behavior on \({\mathbb{R}}^2\) (in preparation)Google Scholar
  28. 28.
    Kaczynski, T., Mischaikow, K., Mrozek, M.: Computational homology. Applied Mathematical Sciences, vol. 157. Springer-Verlag, New York (2004)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.The Institute of Statistical MathematicsTachikawaJapan
  2. 2.Division of Mathematical and Physical SciencesKanazawa UniversityKanazawaJapan
  3. 3.Graduate school of Information SciencesTohoku UniversityAobaJapan

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