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

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Abstract

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.

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References

  1. Konno, N.: Qunatum Walks. Lecture Notes in Mathematics. Springer, Berlin (2008)

    Google Scholar 

  2. Manouchehri, K., Wang, J.: Physical Implementation of Quantum Walks. Springer, Berlin (2014)

    Book  Google Scholar 

  3. Portugal, R.: Quantum Walks and Search Algorithms. Springer, New York (2013)

    Book  Google Scholar 

  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)

  5. Meyer, D.A.: From quantum cellular automata to quantum lattice gases. J. Stat. Phys. 85, 551–574 (1996)

    Article  MathSciNet  ADS  Google Scholar 

  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)

  7. Childs, A.: Universal computation by quantum walk. Phys. Rev. Lett. 102, 180501 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  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)

  9. Kitagawa, T., Rudner, M.S., Berg, E., Demler, E.: Exploring topological phases with quantum walks. Phys. Rev. A 82, 033429 (2010)

    Article  ADS  Google Scholar 

  10. Ahlbrecht, A., Scholz, V.B., Werner, A.H.: Disordered quantum walks in one lattice dimension. J. Math. Phys. 52, 102201 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  12. Miyazaki, T., Konno, M., Konno, N.: Wigner formula of rotation matrices and quantum walks. Phys. Rev. A 76, 012332 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  13. Stefanak, M., Jex, I., Kiss, T.: Recurrence and Polya number of quantum walks. Phys. Rev. Lett. 100, 020501 (2008)

    Article  ADS  Google Scholar 

  14. Godsil, C., Guo, K.: Quantum walks on regular graphs and eigenvalues. Electr. J. Comb. 18, 165 (2011)

    MathSciNet  MATH  Google Scholar 

  15. Konno, N., Mitsuhashi, H., Sato, I.: The quaternionic weighted zeta function of a graph. J. Algebr. Comb. 44, 729–755 (2016)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  ADS  Google Scholar 

  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)

    Article  MathSciNet  ADS  Google Scholar 

  19. Cantero, M.J., Grünbaum, F.A., Moral, L., Velázquez, L.: The CGMV method for quantum walks. Quantum Inf. Process. 11, 1149–1192 (2012)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  ADS  Google Scholar 

  21. Suzuki, A.: Asymptotic velocity of a position dependent quantum walk. Quantum Inf. Process. 15, 103–119 (2016)

    Article  MathSciNet  ADS  Google Scholar 

  22. Feynman, R.F., Hibbs, A.R.: Quantum Mechanics and Path Integrals. McGraw-Hill Inc., New York (1965)

    MATH  Google Scholar 

  23. Higuchi, Yu., Konno, N., Sato, I., Segawa, E.: Quantum graph walks I: mapping to quantum walks. Yokohama Math. J. 59, 33–55 (2013)

    MathSciNet  MATH  Google Scholar 

  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)

  25. Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holeden, H.: Solvable Models in Quantum Mechanics. AMS Chelsea Publishing, Madison (2004)

    Google Scholar 

  26. Exner, P., Seba, P.: Free quantum motion on a branching graph. Rep. Math. Phys. 28, 7–26 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  27. Gnutzmann, S., Smilansky, U.: Quantum graphs: applications to quantum chaos and universal spectral statistics. Adv. Phys. 55, 527–625 (2006)

    Article  ADS  Google Scholar 

  28. Jona-Lasonio, G., Presilla, C., Sjösrand, J.: On Schrödinger equations with concentrated nonlinearities. Anal. Phys. 240, 1–21 (1995)

    Article  ADS  Google Scholar 

  29. Adami, R., Teta, A.: A class of nonlinear Schrodinger equations with concentrated nonlinearity. J. Funct. Anal. 180, 148–175 (2001)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  31. Shikano, Y., Wada, T., Horikawa, J.: Discrete-time quantum walk with feed-forward quantum coin. Sci. Rep. 4, 4427–4434 (2014)

    Article  ADS  Google Scholar 

  32. Konno, N.: Quantum random walks in one dimension. Quantum Inf. Process. 1, 345–354 (2002)

    Article  MathSciNet  Google Scholar 

  33. Konno, N.: A new type of limit theorems for the one-dimensional quantum random walk. J. Math. Soc. Jpn. 57, 1179–1195 (2005)

    Article  MathSciNet  Google Scholar 

  34. Molina, M.I., Bustamante, C.A.: The attractive nonlinear delta-function potential. arXiv:physics/0102053

  35. Adami, R., Cacciapuoti, C., Finco, D., Noja, D.: Fast solitons on star graphs. Rev. Math. Phys. 23, 409–451 (2011)

    Article  MathSciNet  Google Scholar 

  36. Sunada, T., Tate, T.: Asymptotic behavior of quantum walks on the line. J. Funct. Anal. 262, 2608–2645 (2012)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

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.

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Authors and Affiliations

  1. DISMA -Dipartimento di Scienze Matematiche “G. L. Lagrange”, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129, Turin, Italy

    Riccardo Adami

  2. Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai, 980-8579, Japan

    Reika Fukuizumi

  3. Graduate School of Education Center, Yokohama National University, Yokohama, 240-8501, Japan

    Etsuo Segawa

Authors
  1. Riccardo Adami
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  2. Reika Fukuizumi
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  3. Etsuo Segawa
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Correspondence to Etsuo Segawa.

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Adami, R., Fukuizumi, R. & Segawa, E. A nonlinear quantum walk induced by a quantum graph with nonlinear delta potentials. Quantum Inf Process 18, 119 (2019). https://doi.org/10.1007/s11128-019-2215-8

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  • DOI: https://doi.org/10.1007/s11128-019-2215-8

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