Advertisement

Graphs and Combinatorics

, Volume 33, Issue 6, pp 1419–1432 | Cite as

Quaternionic Grover Walks and Zeta Functions of Graphs with Loops

  • Norio Konno
  • Hideo Mitsuhashi
  • Iwao Sato
Original Paper

Abstract

For a graph with at most one loop at each vertex, we define a discrete-time quaternionic quantum walk on the graph, which can be viewed as a quaternionic extension of the Grover walk on the graph. We derive the unitary condition for the transition matrix of the quaternionic Grover walk, and discuss the relationship between the right spectra of the transition matrices and zeta functions of graphs.

Keywords

Quantum walk Ihara zeta function Quaternion 

Notes

Acknowledgements

N. Konno 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). H. Mitsuhashi and I. Sato are partially supported by the Grant-in-Aid for Scientific Research (C) of Japan Society for the Promotion of Science (Grant No. 16K05249 and No. 15K04985, respectively). We thank K. Tamano, Y. Ide, and O. Kada for helpful comments at the stage of the revision process. We would also like to thank the referee for careful reading and for fruitful suggestions.

References

  1. 1.
    Adler, S.L.: Quaternion Quantum Mechanics and Quantum Fields. Oxford University Press, Oxford (1995)zbMATHGoogle Scholar
  2. 2.
    Ambainis, A.: Quantum walks and their algorithmic applications. Int. J. Quantum Inf. 1, 507 (2003)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bass, H.: The Ihara-Selberg zeta function of a tree lattice. Int. J. Math. 3, 717–797 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Birkhoff, G., von Neumann, J.: The Logic of Quantum Mechanics. Ann. Math. 37, 823–843 (1936)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Emms, D., Hancock, E.R., Severini, S., Wilson, R.C.: A matrix representation of graphs and its spectrum as a graph invariant. Electron J. Comb. 13, R34 (2006)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the 28th Annual ACM Symposium on the Theory of Computing (STOC), pp. 212–219 (1996)Google Scholar
  7. 7.
    Hashimoto, K.: Zeta functions of finite graphs and representations of \(p\)-adic groups. Adv. Stud. Pure Math. 15, 211–280 (1989)MathSciNetGoogle Scholar
  8. 8.
    Ihara, Y.: On discrete subgroups of the two by two projective linear group over \(p\)-adic fields. J. Math. Soc. Jpn. 18, 219–235 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Konno, N.: Quantum Walk (in Japanese). Morikita Publishing Co., Ltd. (2014)Google Scholar
  10. 10.
    Konno, N.: Quaternionic quantum walks. Quantum Stud. 2, 63–76 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Konno, N., Sato, I.: On the relation between quantum walks and zeta functions. Quantum Inf. Process. 11, 341–349 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Konno, N., Mitsuhashi, H., Sato, I.: The discrete-time quaternionic quantum walk on a graph. Quantum Inf. Process. 15, 651–673 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Konno, N., Mitsuhashi, H., Sato, I.: The discrete-time quaternionic quantum walk and the second weighted zeta function on a graph. Interdiscip. Inf. Sci. 23(1), 9–17 (2017)MathSciNetGoogle Scholar
  14. 14.
    Manouchehri, K., Wang, J.: Physical Implementation of Quantum Walks. Springer, Berlin (2013)zbMATHGoogle Scholar
  15. 15.
    Mizuno, H., Sato, I.: The scattering matrix of a graph. Electron. J. Comb. 15, R96 (2008)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Portugal, R.: Quantum Walks and Search Algorithms. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  17. 17.
    Sato, I.: A new Bartholdi zeta function of a graph. Int. J. Algebra 1, 269–281 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Smilansky, U.: Quantum chaos on discrete graphs. J. Phys. A 40, F621–F630 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Japan 2017

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of EngineeringYokohama National UniversityHodogaya, YokohamaJapan
  2. 2.Department of Applied Informatics, Faculty of Science and EngineeringHosei UniversityKoganeiJapan
  3. 3.Oyama National College of TechnologyOyamaJapan

Personalised recommendations