Necessary and Sufficient Condition for Quantum Computing

  • Koji Nagata
  • Tadao Nakamura
  • Ahmed Farouk
  • Do Ngoc Diep


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.


Quantum algorithms Quantum computation 



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


  1. 1.
    Rennie, R. (ed.): Oxford dictionary of physics, 7th. Oxford University Press, Oxford (2015)Google Scholar
  2. 2.
    Deutsch, D.: Soc, Proc. Roy. London Ser. A 400, 97 (1985)ADSCrossRefGoogle Scholar
  3. 3.
    Deutsch, D., Jozsa, R.: Proc. Roy. Soc. London Ser. A 439, 553 (1992)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Cleve, R., Ekert, A., Macchiavello, C., Mosca, M.: Proc. Roy. Soc. London Ser. A 454, 339 (1998)ADSCrossRefGoogle Scholar
  5. 5.
    Jones, J.A., Mosca, M.: J. Chem. Phys. 109, 1648 (1998)ADSCrossRefGoogle Scholar
  6. 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)ADSCrossRefGoogle Scholar
  7. 7.
    de Oliveira, A.N., Walborn, S.P., Monken, C.H.: J. Opt. B: Quantum Semiclass. Opt. 7, 288–292 (2005)ADSCrossRefGoogle Scholar
  8. 8.
    Kim, Y.-H.: Rev, Phys. A 67(R), 040301 (2003)CrossRefGoogle Scholar
  9. 9.
    Mohseni, M., Lundeen, J.S., Resch, K.J., Steinberg, A.M.: Phys. Rev. Lett. 91, 187903 (2003)ADSCrossRefGoogle Scholar
  10. 10.
    Tame, M.S., Prevedel, R., Paternostro, M., Böhi, P., Kim, M.S., Zeilinger, A.: Phys. Rev. Lett. 98, 140501 (2007)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Bernstein, E., Vazirani, U.: . In: Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing (STOC ’93), pp 11–20 (1993)Google Scholar
  12. 12.
    Bernstein, E., Vazirani, U.: SIAM J. Comput. 26-5, 1411–1473 (1997)CrossRefGoogle Scholar
  13. 13.
    Nagata, K., Resconi, G., Nakamura, T., Batle, J., Abdalla, S., Farouk, A.: MOJ Ecol Environ Sci 2(1), 00010 (2017)Google Scholar
  14. 14.
    Simon, D.R.: Foundations of computer science. In: Proceedings 35th Annual Symposium on: 116-123, retrieved 2011-06-06 (1994)Google Scholar
  15. 15.
    Shor, P.W.: . In: Proceedings of the 35th IEEE Symposium on Foundations of computer science, p 124 (1994)Google Scholar
  16. 16.
    Grover, L.K.: . In: Proceedings of the twenty-eighth annual ACM symposium on theory of computing, p 212 (1996)Google Scholar
  17. 17.
    Du, J., Shi, M., Zhou, X., Fan, Y., Ye, B.J., Han, R., Wu, J.: Phys. Rev. A 64, 042306 (2001)ADSCrossRefGoogle Scholar
  18. 18.
    Brainis, E., Lamoureux, L.-P., Cerf, N.J., Emplit, P.h., Haelterman, M., Massar, S.: Phys. Rev. Lett. 90, 157902 (2003)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Cross, A.W., Smith, G., Smolin, J.A.: Phys. Rev. A 92, 012327 (2015)ADSCrossRefGoogle Scholar
  20. 20.
    Li, H., Yang, L.: Quantum Inf. Process. 14, 1787 (2015)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Adcock, M.R.A., Hoyer, P., Sanders, B.C.: Quantum Inf. Process. 15, 1361 (2016)ADSMathSciNetCrossRefGoogle Scholar
  22. 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)ADSCrossRefGoogle Scholar
  23. 23.
    Diep, D.N., Giang, D.H., Van Minh, N.: Int. J. Theor. Phys. 56, 1948 (2017)CrossRefGoogle Scholar
  24. 24.
    Jin, W.: Quantum Inf. Process. 15, 65 (2016)ADSMathSciNetCrossRefGoogle Scholar
  25. 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. 26.
    Nagata, K., Nakamura, T., Geurdes, H., Batle, J., Abdalla, S., Farouk, A., Diep, D.N.: Int. J. Theor. Phys. 57, 973 (2018)CrossRefGoogle Scholar
  27. 27.
    Nagata, K., Nakamura, T., Geurdes, H., Batle, J., Abdalla, S., Farouk, A.: Int. J. Theor. Phys. 57, 1605 (2018)CrossRefGoogle Scholar
  28. 28.
    Nagata, K., Nakamura, T.: J Sci Eng Res 5(3), 326–328 (2018)Google Scholar
  29. 29.
    Nagata, K., Nakamura, T., Batle, J., Farouk, A.: Int. J. Theor. Phys. 57, 3098 (2018)CrossRefGoogle Scholar
  30. 30.
    Nagata, K., Nakamura, T.: Open Access Library J. 2, e1798 (2015)Google Scholar
  31. 31.
    Nagata, K., Nakamura, T.: Int. J. Theor. Phys. 56, 2086 (2017)CrossRefGoogle Scholar
  32. 32.
    Nagata, K., Nakamura, T., Farouk, A.: Int. J. Theor. Phys. 56, 2887 (2017)CrossRefGoogle Scholar
  33. 33.
    Diep, D.N., Giang, D.H.: Int. J. Theor. Phys. 56, 2797 (2017)CrossRefGoogle Scholar
  34. 34.
    Diep, D.N., Giang, D.H., Phu, P.H.: Int. J. Theor. Phys. 57, 841 (2018)CrossRefGoogle Scholar
  35. 35.
    Resconi, G., Nagata, K.: Intern. J. Gen. Eng. Technol. 7(1), 1–20 (2018)Google Scholar
  36. 36.
    Nagata, K., Nakamura, T., Geurdes, H., Batle, J., Farouk, A., Diep, D.N., Patro, S.K.: Int. J. Theor. Phys. 57, 2546 (2018)CrossRefGoogle Scholar
  37. 37.
    Nagata, K., Nakamura, T.: Quantum algorithm for the root-finding problem, Quantum Stud.: Math. Found. (2018)
  38. 38.
    Nielsen, M.A., Chuang, I.L.: Quantum computation and quantum information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  39. 39.
    Devitt, S.J., Munro, W.J., Nemoto, K.: Rep. Prog. Phys. 76, 076001 (2013)ADSCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of PhysicsKorea Advanced Institute of Science and TechnologyDaejeonKorea
  2. 2.Department of Information and Computer ScienceKeio UniversityKohoku-ku, YokohamaJapan
  3. 3.Department of Physics and Computer Science, Faculty of ScienceWilfrid Laurier UniversityWaterlooCanada
  4. 4.TIMASThang Long UniversityHanoiVietnam
  5. 5.Institute of Mathematics, VASTHanoiVietnam

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