A nonlinear quantum walk induced by a quantum graph with nonlinear delta potentials

  • Riccardo Adami
  • Reika Fukuizumi
  • Etsuo SegawaEmail author


We study a nonlinear quantum walk naturally induced by a quantum graph with nonlinear delta potentials. We find a strongly ballistic spreading in the behavior of this nonlinear quantum walk with some special initial states.


Nonlinear delta potential Quantum walks Nonlinear quantum graph 



E.S. acknowledges financial supports from the Grant-in-Aid for Young Scientists (B) and of Scientific Research (B) Japan Society for the Promotion of Science (Grant Nos. 16K17637, 16H03939). We would like to thank the referees for helpful comments and suggestions.


  1. 1.
    Konno, N.: Qunatum Walks. Lecture Notes in Mathematics. Springer, Berlin (2008)Google Scholar
  2. 2.
    Manouchehri, K., Wang, J.: Physical Implementation of Quantum Walks. Springer, Berlin (2014)CrossRefGoogle Scholar
  3. 3.
    Portugal, R.: Quantum Walks and Search Algorithms. Springer, New York (2013)CrossRefGoogle Scholar
  4. 4.
    Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J.: One-dimensional quantum walks. In: Proceedings of Symposium on Theory of Computing, pp. 37–49 (2001)Google Scholar
  5. 5.
    Meyer, D.A.: From quantum cellular automata to quantum lattice gases. J. Stat. Phys. 85, 551–574 (1996)MathSciNetCrossRefADSGoogle Scholar
  6. 6.
    Ambainis, A., Kempe, J., Rivosh, A.: Coins make quantum walks faster. In: Proceedings of ACM-SIAM Symposium on Discrete Algorithm, pp. 1099–1108 (2005)Google Scholar
  7. 7.
    Childs, A.: Universal computation by quantum walk. Phys. Rev. Lett. 102, 180501 (2009)MathSciNetCrossRefADSGoogle Scholar
  8. 8.
    Szegedy, M.: Quantum speed-up of Markov chain based algorithms. In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 32–41 (2004)Google Scholar
  9. 9.
    Kitagawa, T., Rudner, M.S., Berg, E., Demler, E.: Exploring topological phases with quantum walks. Phys. Rev. A 82, 033429 (2010)CrossRefADSGoogle Scholar
  10. 10.
    Ahlbrecht, A., Scholz, V.B., Werner, A.H.: Disordered quantum walks in one lattice dimension. J. Math. Phys. 52, 102201 (2011)MathSciNetCrossRefADSGoogle Scholar
  11. 11.
    Oka, T., Konno, N., Arita, R., Aoki, H.: Breakdown of an electric-field driven system: a mapping to a quantum walk. Phys. Rev. Lett. 94, 100602 (2005)CrossRefADSGoogle Scholar
  12. 12.
    Miyazaki, T., Konno, M., Konno, N.: Wigner formula of rotation matrices and quantum walks. Phys. Rev. A 76, 012332 (2007)MathSciNetCrossRefADSGoogle Scholar
  13. 13.
    Stefanak, M., Jex, I., Kiss, T.: Recurrence and Polya number of quantum walks. Phys. Rev. Lett. 100, 020501 (2008)CrossRefADSGoogle Scholar
  14. 14.
    Godsil, C., Guo, K.: Quantum walks on regular graphs and eigenvalues. Electr. J. Comb. 18, 165 (2011)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Konno, N., Mitsuhashi, H., Sato, I.: The quaternionic weighted zeta function of a graph. J. Algebr. Comb. 44, 729–755 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Higuchi, Yu., Konno, N., Sato, I., Segawa, E.: A remark on zeta functions of finite graphs via quantum walks. Pac. J. Math. Ind. 6, 73–84 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Konno, N., Obata, N., Segawa, E.: Localization of the Grover walks on spidernets and free Meixner laws. Commun. Math. Phys. 322, 667–695 (2013)MathSciNetCrossRefADSGoogle Scholar
  18. 18.
    Bourgain, J., Grunbaum, A., Velazquez, L., Wilkening, J.: Quantum recurrence of a subspace and operator-valued Schur functions. Commun. Math. Phys. 329, 1031–1067 (2014)MathSciNetCrossRefADSGoogle Scholar
  19. 19.
    Cantero, M.J., Grünbaum, F.A., Moral, L., Velázquez, L.: The CGMV method for quantum walks. Quantum Inf. Process. 11, 1149–1192 (2012)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Grunbaum, F.A., Velazquez, L., Werner, A.H., Werner, R.F.: Recurrence for discrete time unitary evolutions. Commun. Math. Phys. 320, 543–569 (2013)MathSciNetCrossRefADSGoogle Scholar
  21. 21.
    Suzuki, A.: Asymptotic velocity of a position dependent quantum walk. Quantum Inf. Process. 15, 103–119 (2016)MathSciNetCrossRefADSGoogle Scholar
  22. 22.
    Feynman, R.F., Hibbs, A.R.: Quantum Mechanics and Path Integrals. McGraw-Hill Inc., New York (1965)zbMATHGoogle Scholar
  23. 23.
    Higuchi, Yu., Konno, N., Sato, I., Segawa, E.: Quantum graph walks I: mapping to quantum walks. Yokohama Math. J. 59, 33–55 (2013)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Tanner, G.: From quantum graphs to quantum random walks. Non-Linear Dynamics and Fundamental Interactions, NATO Science Series II: Mathematics, Physics and Chemistry, vol. 213, pp. 69–87 (2006)Google Scholar
  25. 25.
    Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holeden, H.: Solvable Models in Quantum Mechanics. AMS Chelsea Publishing, Madison (2004)Google Scholar
  26. 26.
    Exner, P., Seba, P.: Free quantum motion on a branching graph. Rep. Math. Phys. 28, 7–26 (1989)MathSciNetCrossRefADSGoogle Scholar
  27. 27.
    Gnutzmann, S., Smilansky, U.: Quantum graphs: applications to quantum chaos and universal spectral statistics. Adv. Phys. 55, 527–625 (2006)CrossRefADSGoogle Scholar
  28. 28.
    Jona-Lasonio, G., Presilla, C., Sjösrand, J.: On Schrödinger equations with concentrated nonlinearities. Anal. Phys. 240, 1–21 (1995)CrossRefADSGoogle Scholar
  29. 29.
    Adami, R., Teta, A.: A class of nonlinear Schrodinger equations with concentrated nonlinearity. J. Funct. Anal. 180, 148–175 (2001)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Maeda, M., Sasaki, H., Segawa, E., Suzuki, A., Suzuki, K.: Scattering and inverse scattering for nonlinear quantum walks. Discret. Cont. Dyn. Syst. A 38, 3687–3703 (2018)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Shikano, Y., Wada, T., Horikawa, J.: Discrete-time quantum walk with feed-forward quantum coin. Sci. Rep. 4, 4427–4434 (2014)CrossRefADSGoogle Scholar
  32. 32.
    Konno, N.: Quantum random walks in one dimension. Quantum Inf. Process. 1, 345–354 (2002)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Konno, N.: A new type of limit theorems for the one-dimensional quantum random walk. J. Math. Soc. Jpn. 57, 1179–1195 (2005)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Molina, M.I., Bustamante, C.A.: The attractive nonlinear delta-function potential. arXiv:physics/0102053
  35. 35.
    Adami, R., Cacciapuoti, C., Finco, D., Noja, D.: Fast solitons on star graphs. Rev. Math. Phys. 23, 409–451 (2011)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Sunada, T., Tate, T.: Asymptotic behavior of quantum walks on the line. J. Funct. Anal. 262, 2608–2645 (2012)MathSciNetCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.DISMA -Dipartimento di Scienze Matematiche “G. L. Lagrange”Politecnico di TorinoTurinItaly
  2. 2.Research Center for Pure and Applied Mathematics, Graduate School of Information SciencesTohoku UniversitySendaiJapan
  3. 3.Graduate School of Education CenterYokohama National UniversityYokohamaJapan

Personalised recommendations