Advertisement

Stationary amplitudes of quantum walks on the higher-dimensional integer lattice

  • Takashi Komatsu
  • Norio Konno
Article

Abstract

Stationary measures of quantum walks on the one-dimensional integer lattice are well studied. However, the stationary measure for the higher-dimensional case has not been clarified. In this paper, we give the stationary amplitude for quantum walks on the d-dimensional integer lattice with a finite support by solving the corresponding eigenvalue problem. As a corollary, we can obtain the stationary measures of the Grover walks. In fact, the amplitude for the stationary measure is an eigenfunction with eigenvalue 1.

Keywords

Discrete time quantum walk Stationary amplitude Higher-dimensional integer lattice Stationary measure 

Notes

Acknowledgements

This work is partially supported by the Grant-in-Aid for Scientific Research (Challenging Exploratory Research) of Japan Society for the Promotion of Science (Grant No. 15K13443).

References

  1. 1.
    Aharonov, Y., Davidovich, L., Zagury, N.: Quantum random walks. Phys. Rev. A 48, 1687–1690 (1993)ADSCrossRefGoogle Scholar
  2. 2.
    Ambainis, A., Kempe, J., Rivosh, A.: Coins make quantum walks faster. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, pp. 1099–1108 (2005)Google Scholar
  3. 3.
    Cantero, M.J., Grunbaum, F.A., Moral, L., Velazquez, L.: The CGMV method for quantum walks. Quantum Inf. Process. 11, 1149–1192 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Di Franco, C., Mc Gettrick, M., Machida, T., Busch, T.: Alternate two-dimensional quantum walk with a single-qubit coin. Phys. Rev. A 84, 042337 (2011)ADSCrossRefGoogle Scholar
  5. 5.
    Endo, S., Endo, T., Konno, N., Segawa, E., Takei, M.: Limit theorems of a two-phase quantum walk with one defect. Quantum Inf. Comput. 15, 1373–1396 (2015)MathSciNetGoogle Scholar
  6. 6.
    Endo, T., Kawai, H., Konno, N.: The stationary measure for diagonal quantum walk with one defect, arXiv:1603.08948 (2016)
  7. 7.
    Endo, T., Kawai, H., Konno, N.: Stationary measures for the three-state Grover walk with one defect in one dimension. RIMS Kokyuroku 2010, 45–55 (2016)Google Scholar
  8. 8.
    Endo, T., Konno, N.: The stationary measure of a space-inhomogeneous quantum walk on the line. Yokohama Math. J. 60, 33–47 (2014)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Endo, T., Konno N., Obuse, H.: Relation between two-phase quantum walks and the topological invariant. arXiv:1511.04230 (2015)
  10. 10.
    Endo, T., Konno, N., Segawa, E., Takei, M.: A one-dimensional Hadamard walk with one defect. Yokohama Math. J. 60, 49–90 (2014)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Grimmett, G., Janson, S., Scudo, P.F.: Weak limits for quantum random walks. Phys. Rev. E 69, 026119 (2004)ADSCrossRefGoogle Scholar
  12. 12.
    Higuchi, Y., Konno, N., Sato, I., Segawa, E.: Spectral and asymptotic properties of Grover walks on crystal lattices. J. Funct. Anal. 267, 4197–4235 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Inui, N., Konishi, Y., Konno, N.: Localization of two-dimensional quantum walks. Phys. Rev. A 69, 052323 (2004)ADSCrossRefGoogle Scholar
  14. 14.
    Inui, N., Konno, N., Segawa, E.: One-dimensional three-state quantum walk. Phys. Rev. E 72, 056112 (2005)ADSCrossRefGoogle Scholar
  15. 15.
    Kawai, H., Komatsu, T., Konno, N.: Stationary measures of three-state quantum walks on the one-dimensional lattice. arXiv:1702.01523 (2017)
  16. 16.
    Komatsu, T.: Limiting distributions of quantum walks on the square lattice. Yokohama Math. J. 61, 67–86 (2015)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Komatsu, T., Tate, T.: Eigenvalues of quantum walks of Grover and Fourier types. arXiv:1704.05236 (2017)
  18. 18.
    Konno, N.: A new type of limit theorems for the one-dimensional quantum random walk. J. Math. Soc. Jpn. 57, 1179–1195 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Konno, N.: The uniform measure for discrete-time quantum walks in one dimension. Quantum Inf. Process. 13, 1103–1125 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Konno, N., Luczak, T., Segawa, E.: Limit measures of inhomogeneous discrete-time quantum walks in one dimension. Quantum Inf. Process. 12, 33–53 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Konno, N., Takei, M.: The non-uniform stationary measure for discrete-time quantum walks in one dimension. Quantum Inf. Comput. 15, 1060–1075 (2015)MathSciNetGoogle Scholar
  22. 22.
    Machida, T., Chandrashekar, C.M., Konno, N., Busch, T.: Limit distributions for different forms of four-state quantum walks on a two-dimensional lattice. Quantum Inf. Comput. 15, 1248–1258 (2015)MathSciNetGoogle Scholar
  23. 23.
    Manouchehri, K., Wang, J.: Physical Implementation of Quantum Walks. Springer, Berlin (2013)zbMATHGoogle Scholar
  24. 24.
    Portugal, R.: Quantum Walks and Search Algorithms. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  25. 25.
    Tate, T.: Eigenvalues, absolute continuity and localizations for periodic unitary transition operators. arXiv:1411.4215 (2014)
  26. 26.
    Wang, C., Lu, X., Wang, W.: The stationary measure of a space-inhomogeneous three-state quantum walk on the line. Quantum Inf. Process. 14, 867–880 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Watabe, K., Kobayashi, N., Katori, M., Konno, N.: Limit distributions of two-dimensional quantum walks. Phys. Rev. A 77, 062331 (2008)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of EngineeringYokohama National UniversityYokohamaJapan

Personalised recommendations