Advertisement

Discrete-time quantum walk search on Johnson graphs

  • Xi-ling XueEmail author
  • Yue Ruan
  • Zhi-hao Liu
Article

Abstract

The Johnson graph J (n, k) is defined by n symbols, where vertices are k-element subsets of the symbols, and vertices are adjacent if they differ in exactly one symbol. In particular, both J (n, 1), the complete graph Kn, and J (n, 2), the strongly regular triangular graph Tn, support fast quantum spatial search. Wong showed that continuous-time quantum walk search on J (n, 3) also supports fast search. The problem is reconsidered in the language of scattering quantum walk, a type of discrete-time quantum walk. Here the search space is confined to a low-dimensional subspace corresponding to the collapsed graph. Using matrix perturbation theory, we show that discrete-time quantum walk search on J (n, 3) also achieves full quantum speedup. The analytical method can also be applied to general Johnson graphs J (n, k) with fixed k.

Keywords

Discrete-time quantum walks Quantum spatial search Johnson graphs 

Notes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 61802002, 61502101) and the Natural Science Foundation of Anhui Province, China (Grant No. 1708085MF162).

References

  1. 1.
    Childs, A.M.: Universal computation by quantum walk. Phys. Rev. Lett. 102, 180501 (2009)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Childs, A.M., Gosset, D., Webb, Z.: Universal computation by multiparticle quantum walk. Science 339(6121), 791–794 (2013)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Lovett, N.B., Cooper, S., Everitt, M., Kendon, V.: Universal quantum computation using the discrete-time quantum walk. Phys. Rev. A 81, 042330 (2010)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Dheeraj, M.N., Brun, T.A.: Continuous limit of discrete quantum walks. Phys. Rev. A 91, 6 (2015)Google Scholar
  5. 5.
    Shenvi, N., Kempe, J., Whaley, K.B.: Quantum random-walk search algorithm. Phys. Rev. A 67, 052307 (2003)ADSCrossRefGoogle Scholar
  6. 6.
    Ambainis, A., Kempe, J., Rivosh, A.: Coins make quantum walks faster. In: Proceedings of 16th ACM-SIAM SODA, pp. 1099–1108 (2005)Google Scholar
  7. 7.
    Reitzner, D., Hillery, M., Feldman, E.: Quantum searches on highly symmetric graphs. Phys. Rev. A 79, 012323 (2009)ADSCrossRefGoogle Scholar
  8. 8.
    Hillery, M., Reitzner, D., Bužek, V.: Searching via walking: how to find a marked clique of a complete graph using quantum walks. Phys. Rev. A 81, 062324 (2010)ADSCrossRefGoogle Scholar
  9. 9.
    Janmark, J., Meyer, D.A., Wong, T.G.: Global symmetry is unnecessary for fast quantum search. Phys. Rev. Lett. 112, 210502 (2014)ADSCrossRefGoogle Scholar
  10. 10.
    Reitzner, D., Nagaj, D., Buzek, V.: Quantum walks. Acta Phys. Slovaca 61, 6 (2011)CrossRefGoogle Scholar
  11. 11.
    Babai, L.: Graph isomorphism in quasipolynomial time. arXiv:1512.03547 (2015)
  12. 12.
    Ambainis, A.: Quantum walk algorithm for element distinctness. SIAM J. Comput. 37(1), 210–239 (2007)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Cao, W.F., Zhang, Y.C., Yang, Y.G., et al.: Constructing quantum Hash functions based on quantum walks on Johnson graphs. Quantum Inf. Process. 17(7), 156 (2018)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Wong, T.G.: Quantum walk search on Johnson graphs. J. Phys. A Math. Theor. 49(19), 195303 (2016)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Bose, R.L.R.C.: A characterization of tetrahedral graphs. J. Comb. Theory 3, 366–385 (1967)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Xue, X.L., Liu, Z.H., Chen, H.W.: Search algorithm on strongly regular graphs based on scattering quantum walk. Chin. Phys. B 1, 108–114 (2017)Google Scholar
  17. 17.
    Cottrell, S.S.: Finding structural anomalies in star graphs using quantum walks: a general approach. J. Phys. A Math. Theor. 48, 035304 (2015)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Grover, L.K.: Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 79, 325 (1997)ADSCrossRefGoogle Scholar
  19. 19.
    Hillery, M., Zheng, H., Feldman, E., et al.: Quantum walks as a probe of structural anomalies in graphs. Phys. Rev. A 85(6), 062325 (2012)ADSCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.School of Computer Science and TechnologyAnhui University of TechnologyMaanshanChina
  2. 2.School of Computer Science and EngineeringSoutheast UniversityNanjingChina

Personalised recommendations