Exact simulation of coined quantum walks with the continuous-time model



The connection between coined and continuous-time quantum walk models has been addressed in a number of papers. In most of those studies, the continuous-time model is derived from coined quantum walks by employing dimensional reduction and taking appropriate limits. In this work, we produce the evolution of a coined quantum walk on a generic graph using a continuous-time quantum walk on a larger graph. In addition to expanding the underlying structure, we also have to switch on and off edges during the continuous-time evolution to accommodate the alternation between the shift and coin operators from the coined model. In one particular case, the connection is very natural, and the continuous-time quantum walk that simulates the coined quantum walk is driven by the graph Laplacian on the dynamically changing expanded graph.


Quantum walks Quantum walk on graphs Coined quantum walk Continuous-time quantum walk 



P.P. would like to thank CNPq for its financial support (Grant Nos. 400216/2014-0 and 150726/2015-5). R.P. acknowledges financial support from Faperj (Grant No. E-26/102.350/2013) and CNPq (Grant Nos. 303406/2015-1 and 474143/2013-9).


  1. 1.
    Aharonov, Y., Davidovich, L., Zagury, N.: Quantum random walks. Phys. Rev. A 48, 1687–1690 (1993)ADSCrossRefGoogle Scholar
  2. 2.
    Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: Quantum walks on graphs. In: Proceedings of the Thirty-third Annual ACM Symposium on Theory of Computing, STOC ’01, pp. 50–59. ACM, New York (2001)Google Scholar
  3. 3.
    Farhi, E., Gutmann, S.: Quantum computation and decision trees. Phys. Rev. A 58, 915–928 (1998)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Kempe, J.: Quantum random walks: an introductory overview. Contemp. Phys. 44(4), 307–327 (2003)ADSCrossRefGoogle Scholar
  5. 5.
    Hillery, M., Bergou, J., Feldman, E.: Quantum walks based on an interferometric analogy. Phys. Rev. A 68, 032314 (2003)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Hughes, B.D.: Random Walks and Random Environments: Random walks, vol. 1. Clarendon Press, Oxford (1995)MATHGoogle Scholar
  7. 7.
    Lawler, G.F., Limic, V.: Random Walk: A Modern Introduction. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2010)CrossRefMATHGoogle Scholar
  8. 8.
    Ambainis, A.: Quantum walks and their algorithmic applications. Int. J. Quantum Inf. 01(04), 507–518 (2003)CrossRefMATHGoogle Scholar
  9. 9.
    Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the Twenty-eighth Annual ACM Symposium on Theory of Computing, STOC ’96, pp. 212–219. ACM, New York (1996)Google Scholar
  10. 10.
    Childs, A.M., Goldstone, J.: Spatial search by quantum walk. Phys. Rev. A 70, 022314 (2004)ADSCrossRefGoogle Scholar
  11. 11.
    Ambainis, A., Kempe, J., Rivosh, A.: Coins make quantum walks faster. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’05, pp. 1099–1108. Society for Industrial and Applied Mathematics, Philadelphia (2005)Google Scholar
  12. 12.
    Wong, T.G.: Faster quantum walk search on a weighted graph. Phys. Rev. A 92, 032320 (2015)ADSCrossRefGoogle Scholar
  13. 13.
    Chakraborty, S., Novo, L., Ambainis, A., Omar, Y.: Spatial search by quantum walk is optimal for almost all graphs. Phys. Rev. Lett. 116, 100501 (2016)ADSCrossRefGoogle Scholar
  14. 14.
    Childs, A.M.: On the relationship between continuous- and discrete-time quantum walk. Commun. Math. Phys. 294(2), 581–603 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Paparo, G.D., Müller, M., Francesc, C., Martin-Delgado, M.A.: Quantum google in a complex network. Sci. Rep. 3, 2773 (2013)ADSCrossRefGoogle Scholar
  16. 16.
    Mallick, A., Mandal, S., Chandrashekar, C.M.: Simulation of neutrino oscillations using discrete-time quantum walk. ArXiv e-prints, April (2016)Google Scholar
  17. 17.
    Meyer, D.A.: On the absence of homogeneous scalar unitary cellular automata. Phys. Lett. A 223(5), 337–340 (1996)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Portugal, R., Santos, R.A.M., Fernandes, T.D., Gonçalves, D.N.: The staggered quantum walk model. Quantum Inf. Process. 15(1), 85–101 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Portugal, R.: Staggered quantum walks on graphs. Phys. Rev. A 93, 062335 (2016)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Strauch, F.W.: Connecting the discrete- and continuous-time quantum walks. Phys. Rev. A 74, 030301 (2006)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    D’Alessandro, D.: Connection between continuous and discrete time quantum walks. From D-dimensional lattices to general graphs. Rep. Math. Phys. 66(1), 85–102 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    di Molfetta, G., Debbasch, F.: Discrete-time quantum walks: continuous limit and symmetries. J. Math. Phys. 53(12), 123302 (2012)Google Scholar
  23. 23.
    Dheeraj, M.N., Brun, T.A.: Continuous limit of discrete quantum walks. Phys. Rev. A 91, 062304 (2015)CrossRefGoogle Scholar
  24. 24.
    Szegedy, M.: Quantum speed-up of Markov chain based algorithms. In: 45th Annual IEEE symposium on foundations of computer science, 2004. Proceedings, pp. 32–41, Oct (2004)Google Scholar
  25. 25.
    Romanelli, A., Siri, R., Abal, G., Auyuanet, A., Donangelo, R.: Decoherence in the quantum walk on the line. Phys. A 347(C), 137–152 (2005)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Oliveira, A.C., Portugal, R., Donangelo, R.: Decoherence in two-dimensional quantum walks. Phys. Rev. A 74(012312), 012312 (2006)Google Scholar
  27. 27.
    Kollár, B., Kiss, T., Novotný, J., Jex, I.: Asymptotic dynamics of coined quantum walks on percolation graphs. Phys. Rev. Lett. 108, 230505 (2012)ADSCrossRefGoogle Scholar
  28. 28.
    Mülken, O., Pernice, V., Blumen, A.: Quantum transport on small-world networks: a continuous-time quantum walk approach. Phys. Rev. E 76, 051125 (2007)ADSCrossRefGoogle Scholar
  29. 29.
    Anishchenko, A., Blumen, A., Mülken, O.: Enhancing the spreading of quantum walks on star graphs by additional bonds. Quantum Inf. Process. 11(5), 1273–1286 (2012)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Darázs, Z., Kiss, T.: Time evolution of continuous-time quantum walks on dynamical percolation graphs. J. Phys. A: Math. Theor. 46(37), 375305 (2013)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Portugal, R.: Establishing the equivalence between Szegedy’s and coined quantum walks using the staggered model. Quantum Inf. Process. 15(4), 1387–1409 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Shapir, Y., Aharony, A., Harris, A.B.: Localization and quantum percolation. Phys. Rev. Lett. 49, 486–489 (1982)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Santos, R.A.M., Portugal, R., Fragoso, M.D.: Decoherence in quantum Markov chains. Quantum Inf. Process. 13(2), 559–572 (2014)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.National Laboratory of Scientific ComputingPetrópolisBrazil

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