Skip to main content

Necessary and Sufficient Condition for Quantum Computing

Abstract

A necessary and sufficient condition for quantum computing performed with, for example, the Deutsch-Jozsa algorithm or the Bernstein-Vazirani algorithm, has theoretically been investigated. Assume a 2N qubit-quantum computing which starts with the state \(|\overbrace {0,0,...,0,1}^{N}\rangle |\overbrace {1,1,...,1}^{N}\rangle \) as follows: Uf|0,0,...,0,1〉|1,1,...,1〉 = |0,0,...,0,1〉 \( |\overline {f(0,0,...,0,1)}\rangle . \) Surprisingly the relation f(x) = f(−x) is the necessary and sufficient condition of holding this fundamental relation if local unitary operations can be used.

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

References

  1. Rennie, R. (ed.): Oxford dictionary of physics, 7th. Oxford University Press, Oxford (2015)

  2. Deutsch, D.: Soc, Proc. Roy. London Ser. A 400, 97 (1985)

    ADS  Article  Google Scholar 

  3. Deutsch, D., Jozsa, R.: Proc. Roy. Soc. London Ser. A 439, 553 (1992)

    ADS  MathSciNet  Article  Google Scholar 

  4. Cleve, R., Ekert, A., Macchiavello, C., Mosca, M.: Proc. Roy. Soc. London Ser. A 454, 339 (1998)

    ADS  Article  Google Scholar 

  5. Jones, J.A., Mosca, M.: J. Chem. Phys. 109, 1648 (1998)

    ADS  Article  Google Scholar 

  6. Gulde, S., Riebe, M., Lancaster, G.P.T., Becher, C., Eschner, J., Häffner, H., Schmidt-Kaler, F., Chuang, I.L., Blatt, R.: Nat. (London) 421, 48 (2003)

    ADS  Article  Google Scholar 

  7. de Oliveira, A.N., Walborn, S.P., Monken, C.H.: J. Opt. B: Quantum Semiclass. Opt. 7, 288–292 (2005)

  8. Kim, Y.-H.: Rev, Phys. A 67(R), 040301 (2003)

    Article  Google Scholar 

  9. Mohseni, M., Lundeen, J.S., Resch, K.J., Steinberg, A.M.: Phys. Rev. Lett. 91, 187903 (2003)

    ADS  Article  Google Scholar 

  10. Tame, M.S., Prevedel, R., Paternostro, M., Böhi, P., Kim, M.S., Zeilinger, A.: Phys. Rev. Lett. 98, 140501 (2007)

    ADS  MathSciNet  Article  Google Scholar 

  11. Bernstein, E., Vazirani, U.: . In: Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing (STOC ’93), pp 11–20 (1993)

  12. Bernstein, E., Vazirani, U.: SIAM J. Comput. 26-5, 1411–1473 (1997)

    Article  Google Scholar 

  13. Nagata, K., Resconi, G., Nakamura, T., Batle, J., Abdalla, S., Farouk, A.: MOJ Ecol Environ Sci 2(1), 00010 (2017)

    Google Scholar 

  14. Simon, D.R.: Foundations of computer science. In: Proceedings 35th Annual Symposium on: 116-123, retrieved 2011-06-06 (1994)

  15. Shor, P.W.: . In: Proceedings of the 35th IEEE Symposium on Foundations of computer science, p 124 (1994)

  16. Grover, L.K.: . In: Proceedings of the twenty-eighth annual ACM symposium on theory of computing, p 212 (1996)

  17. Du, J., Shi, M., Zhou, X., Fan, Y., Ye, B.J., Han, R., Wu, J.: Phys. Rev. A 64, 042306 (2001)

    ADS  Article  Google Scholar 

  18. Brainis, E., Lamoureux, L.-P., Cerf, N.J., Emplit, P.h., Haelterman, M., Massar, S.: Phys. Rev. Lett. 90, 157902 (2003)

    ADS  MathSciNet  Article  Google Scholar 

  19. Cross, A.W., Smith, G., Smolin, J.A.: Phys. Rev. A 92, 012327 (2015)

    ADS  Article  Google Scholar 

  20. Li, H., Yang, L.: Quantum Inf. Process. 14, 1787 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  21. Adcock, M.R.A., Hoyer, P., Sanders, B.C.: Quantum Inf. Process. 15, 1361 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  22. Fallek, S.D., Herold, C.D., McMahon, B.J., Maller, K.M., Brown, K.R., Amini, J.M.: New J. Phys. 18, 083030 (2016)

    ADS  Article  Google Scholar 

  23. Diep, D.N., Giang, D.H., Van Minh, N.: Int. J. Theor. Phys. 56, 1948 (2017)

    Article  Google Scholar 

  24. Jin, W.: Quantum Inf. Process. 15, 65 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  25. Nagata, K., Resconi, G., Nakamura, T., Batle, J., Abdalla, S., Farouk, A., Geurdes, H.: Asian J. Math. Phys. 1(1), 1–4 (2017)

    Google Scholar 

  26. Nagata, K., Nakamura, T., Geurdes, H., Batle, J., Abdalla, S., Farouk, A., Diep, D.N.: Int. J. Theor. Phys. 57, 973 (2018)

    Article  Google Scholar 

  27. Nagata, K., Nakamura, T., Geurdes, H., Batle, J., Abdalla, S., Farouk, A.: Int. J. Theor. Phys. 57, 1605 (2018)

    Article  Google Scholar 

  28. Nagata, K., Nakamura, T.: J Sci Eng Res 5(3), 326–328 (2018)

    Google Scholar 

  29. Nagata, K., Nakamura, T., Batle, J., Farouk, A.: Int. J. Theor. Phys. 57, 3098 (2018)

    Article  Google Scholar 

  30. Nagata, K., Nakamura, T.: Open Access Library J. 2, e1798 (2015)

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  33. Diep, D.N., Giang, D.H.: Int. J. Theor. Phys. 56, 2797 (2017)

    Article  Google Scholar 

  34. Diep, D.N., Giang, D.H., Phu, P.H.: Int. J. Theor. Phys. 57, 841 (2018)

    Article  Google Scholar 

  35. Resconi, G., Nagata, K.: Intern. J. Gen. Eng. Technol. 7(1), 1–20 (2018)

    Google Scholar 

  36. Nagata, K., Nakamura, T., Geurdes, H., Batle, J., Farouk, A., Diep, D.N., Patro, S.K.: Int. J. Theor. Phys. 57, 2546 (2018)

    Article  Google Scholar 

  37. Nagata, K., Nakamura, T.: Quantum algorithm for the root-finding problem, Quantum Stud.: Math. Found. https://doi.org/10.1007/s40509-018-0171-0 (2018)

  38. Nielsen, M.A., Chuang, I.L.: Quantum computation and quantum information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  39. Devitt, S.J., Munro, W.J., Nemoto, K.: Rep. Prog. Phys. 76, 076001 (2013)

    ADS  Article  Google Scholar 

Download references

Acknowledgements

We thank Professor Han Geurdes, Professor Shahrokh Heidari, Professor Hamed Daei Kasmaei, and Professor Mark Behzad Doost for valuable comments.

Author information

Authors and Affiliations

Authors

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., Farouk, A. et al. Necessary and Sufficient Condition for Quantum Computing. Int J Theor Phys 58, 136–142 (2019). https://doi.org/10.1007/s10773-018-3917-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-018-3917-x

Keywords

  • Quantum algorithms
  • Quantum computation