Quantum Information Processing

, Volume 10, Issue 3, pp 405–417 | Cite as

Quantum walks, Ihara zeta functions and cospectrality in regular graphs

  • Peng RenEmail author
  • Tatjana Aleksić
  • David Emms
  • Richard C. Wilson
  • Edwin R. Hancock


In this paper we explore an interesting relationship between discrete-time quantum walks and the Ihara zeta function of a graph. The paper commences by reviewing the related literature on the discrete-time quantum walks and the Ihara zeta function. Mathematical definitions of the two concepts are then provided, followed by analyzing the relationship between them. Based on this analysis we are able to account for why the Ihara zeta function can not distinguish cospectral regular graphs. This analysis suggests a means by which to develop zeta functions that have potential in distinguishing such structures.


Quantum walks Ihara zeta functions Cospectrality 


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  1. 1.
    Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: Quantum walks on graphs. In: STOC’01: Proceedings of ACM Theory of Computing, pp. 50–59. ACM Press, New York (2001)Google Scholar
  2. 2.
    Ambainis A.: Quantum walks and their algorithmic applications. Int. J. Quantum Inf. 1, 507–518 (2003)CrossRefzbMATHGoogle Scholar
  3. 3.
    Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J.: One-dimensional quantum walks. In: Proceedings of 33th STOC, pp. 60–69. New York, NY, ACM, NewYork (2001)Google Scholar
  4. 4.
    Bass H.: The Ihara-Selberg zeta function of a tree Lattice. Int’l J. Math. 6, 717–797 (1992)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Cameron P.J.: Strongly Regular Graphs. Topics in Algebraic Graph Theory, pp. 203–221. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  6. 6.
    Childs A.M., Farhi E., Gutmann S.: An example of the difference between quantum and classical random walks. Quantum Inf. Process. 1(1/2), 35–43 (2002)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Childs A.M.: Universal computation by quantum walk. Phys. Rev. Lett. 102(18), 180501 (2009)PubMedCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Douglas B.L., Wang J.B.: A classical approach to the graph isomorphism problem using quantum walks. J. Phys. A Math. Theor. 41(7), 075303 (2008)CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Emms, D., Hancock, E.R., Severini, S., Wilson, R.C.: A matrix representation of graphs and its spectrum as a graph invariant, Electronic J. Combinatorics 13(R34), (2006)Google Scholar
  10. 10.
    Emms, D.: Analysis of Graph Structure Using Quantum Walks, Ph.D. Thesis, University of York (2008)Google Scholar
  11. 11.
    Emms D., Severini S., Wilson R.C., Hancock E.R.: Coined quantum walks lift the cospectrality of graphs and trees. Pattern Recognit. 42(9), 1988–2002 (2009)CrossRefzbMATHGoogle Scholar
  12. 12.
    Gamble J.K., Friesen M., Zhou D., Joynt R., Coppersmith S.N.: Two-particle quantum walks applied to the graph isomorphism problem. Phys. Rev. A 81(5), 52313 (2010)CrossRefADSGoogle Scholar
  13. 13.
    Grover, L.: A fast quantum mechanical algorithm for database search. In: Proceedings of the 28th Annual ACM Symposium on the Theory of Computation, pp. 212–219. New York, NY, ACM Press, New York (1996)Google Scholar
  14. 14.
    Hashimoto K.: Artin-type L-functions and the density theorem for prime cycles on finite graphs. Adv. Stud. Pure Math. 15, 211–280 (1989)Google Scholar
  15. 15.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Kempe J.: Quantum random walks—an introductory overview. Contemp. Phys. 44(4), 307–327 (2003)CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Konno N.: One-dimensional discrete-time quantum walks on random environments. Quantum Inf. Process. 8(5), 387–399 (2009)CrossRefzbMATHADSMathSciNetGoogle Scholar
  18. 18.
    Kotani M., Sunada T.: Zeta functions of finite graphs. J. Math. Sci. Univ. Tokyo 7(1), 7–25 (2000)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Scott G., Storm C.: The coefficients of the Ihara zeta function. Involve A J. Math. 1(2), 217–233 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Shankar R.: Principles of Quantum Mechanics. 2nd edn. Plenum, New York (1994)zbMATHGoogle Scholar
  21. 21.
    Shiau S.-Y., Joynt R., Coppersmith S.N.: Physically-motivated dynamical algorithms for the graph isomorphism problem. Quantum Inform. Comput. 5(6), 492–506 (2005)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Stark H.M., Terras A.A.: Zeta functions of finite graphs and coverings. Adv. Math. 121, 124–165 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Stark H.M., Terras A.A.: Zeta functions of finite graphs and coverings, II. Adv. Math. 154, 132–195 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Stark H.M., Terras A.A.: Zeta functions of finite graphs and coverings, III. Adv. Math. 208(2), 467–489 (2007)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Peng Ren
    • 1
    Email author
  • Tatjana Aleksić
    • 2
  • David Emms
    • 1
  • Richard C. Wilson
    • 1
  • Edwin R. Hancock
    • 1
  1. 1.Department of Computer ScienceThe University of YorkYorkUK
  2. 2.Faculty of ScienceUniversity of KragujevacKragujevacSerbia

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