Skip to main content

Efficient Quantum Algorithm for the Parity Problem of a Certain Function


Based on a particular mathematical structure of a certain function f(x) under our attention, we present a novel quantum algorithm. The algorithm allows one to determine the property of a certain function. In our study, it is f(x) = f(−x). Therefore, there would be a question here, “How fast can we succeed in this?” All we need to do is only the evaluation of a single quantum state \(|\overbrace {0,0,\ldots ,0,1}^{N}\rangle \) (N ≥ 2). Only using that with a little amount of information, we can derive the global property f(x) = f(−x). Our quantum algorithm overcomes a classical counterpart by a factor of the order of 2N.

This is a preview of subscription content, access via your institution.


  1. Lo, H.-K., Popescu, S., Spiller, T. (eds.): Introduction to Quantum Computation and Information. World Scientific, Singapore (1998)

  2. Galindo, A., Martín-Delgado, M.A.: Rev. Mod Phys. 74, 347 (2002)

    ADS  Article  Google Scholar 

  3. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  4. Williams, C.P., Clearwater, S.H.: Explorations in Quantum Computing. Springer, New York (1997)

    MATH  Google Scholar 

  5. Williams, C.P. (ed.): Quantum Computing and Quantum Communications. Springer, Berlin (1998)

  6. Ekert, A.K.: Phys. Rev. Lett. 67, 661 (1991)

    ADS  MathSciNet  Article  Google Scholar 

  7. Bennett, C.H., Brassard, G., Crepeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Phys. Rev. Lett. 70, 1895 (1993)

    ADS  MathSciNet  Article  Google Scholar 

  8. Bennett, C.H., Wiesner, S.J.: Phys. Rev. Lett. 69, 2881 (1993)

    ADS  Article  Google Scholar 

  9. Ekert, A., Jozsa, R.: Rev. Mod. Phys. 68, 773 (1996)

    ADS  Article  Google Scholar 

  10. Berman, G.P., Doolen, G.D., Mainieri, R., Tsifrinovich, I.: Introduction to Quantum Computers. World Scientific, Singapore (1998)

    Book  MATH  Google Scholar 

  11. Feynman, R.: Int. J. Theor. Phys. 21, 467 (1982)

    Article  Google Scholar 

  12. Bernstein, E., Vazirani, U: Proceedings of 25th Annual ACM Symposium on Theory of Computing (STOC ’93), pp. 11. 1993)

  13. Bernstein, E., Vazirani, U.: SIAM J. Comput. 26, 1411 (1997)

    MathSciNet  Article  Google Scholar 

  14. Deutsch, D.: Proc. R. Soc. Lond. A 400, 97 (1985)

    ADS  Article  Google Scholar 

  15. Shor, P.W.: Proceedings of 35th IEEE Annual Symposium on Foundations of Computer Science, pp. 124 (1994)

  16. Grover, L.K.: Proceedings of 28th Annual ACM Symposium on Theory of Computing, pp. 212 (1996)

  17. Smith, J., Mosca, M.: Handbook of natural computing, pp. 1451 (2012)

  18. Childs, A.M., Van Dam, W.: Rev. Mod. Phys. 82, 1 (2010)

    ADS  Article  Google Scholar 

  19. Deutsch, D., Jozsa, R.: Proc. R. Soc. Lond. A 439, 553 (1992)

    ADS  Article  Google Scholar 

  20. Cleve, R., Ekert, A., Macchiavello, C., Mosca, M.: Proc. R. Soc. Lond. A 454, 339 (1998)

    ADS  Article  Google Scholar 

  21. Nagata, K., Nakamura, T.: Open Access Libr. J. 2, e1798 (2015)

    Google Scholar 

  22. Nagata, K., Nakamura, T.: Int. J. Theor. Phys. 56, 2086 (2017)

    Article  Google Scholar 

  23. Nagata, K., Nakamura, T., Farouk, A.: Int. J. Theor. Phys. 56, 2887 (2017)

    Article  Google Scholar 

Download references


We thank Professor Han Geurdes for valuable discussions.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Koji Nagata.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Nagata, K., Nakamura, T., Batle, J. et al. Efficient Quantum Algorithm for the Parity Problem of a Certain Function. Int J Theor Phys 57, 3098–3103 (2018).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Quantum algorithms
  • Quantum computation