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Generalization of the Bernstein–Vazirani algorithm beyond qubit systems

Abstract

Here, we generalize the Bernstein–Vazirani algorithm beyond qubit systems. First, we review the Bernstein–Vazirani algorithm for determining a bit string. Second, we discuss the generalized Bernstein–Vazirani algorithm for determining a natural number string. The speed of determining the strings is shown to outperform the best classical case by a factor of the number of the systems in every cases.

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References

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

  2. Bernstein, E., Vazirani, U.: Quantum complexity theory. SIAM J. Comput. 26–5, 1411–1473 (1997)

    MathSciNet  Article  Google Scholar 

  3. Du, J., Shi, M., Zhou, X., Fan, Y., Ye, B.J., Han, R., Wu, J.: Implementation of a quantum algorithm to solve the Bernstein-Vazirani parity problem without entanglement on an ensemble quantum computer. Phys. Rev. A 64, 042306 (2001)

    Article  Google Scholar 

  4. Brainis, E., Lamoureux, L.-P., Cerf, N.J., Emplit, Ph, Haelterman, M., Massar, S.: Fiber-optics implementation of the Deutsch-Jozsa and Bernstein-Vazirani quantum algorithms with three qubits. Phys. Rev. Lett. 90, 157902 (2003)

    MathSciNet  Article  Google Scholar 

  5. Cross, A.W., Smith, G., Smolin, J.A.: Quantum learning robust against noise. Phys. Rev. A 92, 012327 (2015)

    Article  Google Scholar 

  6. Li, H., Yang, L.: A quantum algorithm for approximating the influences of Boolean functions and its applications Quantum. Inf. Process. 14, 1787 (2015)

    MathSciNet  Article  Google Scholar 

  7. Nagata, K., Nakamura, T.: The Deutsch-Jozsa algorithm can be used for quantum key distribution. Open Access Libr. J. 2, e1798 (2015)

    Google Scholar 

  8. Nagata, K., Nakamura, T., Farouk, A.: Quantum cryptography based on the Deutsch-Jozsa algorithm. Int. J. Theor. Phys. 56, 2887 (2017)

    MathSciNet  Article  Google Scholar 

  9. Fallek, S.D., Herold, C.D., McMahon, B.J., Maller, K.M., Brown, K.R., Amini, J.M.: Transport implementation of the Bernstein-Vazirani algorithm with ion qubits. New. J. Phys. 18, 083030 (2016)

    Article  Google Scholar 

  10. Krishna, R., Makwana, V., Suresh, A.P.: A generalization of Bernstein-Vazirani algorithm to qudit systems. arXiv:1609.03185 [quant-ph] (2016)

  11. Nagata, K., Resconi, G., Nakamura, T., Batle, J., Abdalla, S., Farouk, A.: A generalization of the Bernstein-Vazirani algorithm. MOJ Ecol. Environ. Sci. 2(1), 00010 (2017)

    Google Scholar 

  12. Nagata, K., Nakamura, T., Geurdes, H., Batle, J., Abdalla, S., Farouk, A.: New method of calculating a multiplication by using the generalized Bernstein-Vazirani algorithm. Int. J. Theor. Phys. 57, 1605 (2018)

    MathSciNet  Article  Google Scholar 

  13. Nagata, K., Nakamura, T., Geurdes, H., Batle, J., Farouk, A., Diep, D.N., Patro, S.K.: Efficient quantum algorithms of finding the roots of a polynomial function. Int. J. Theor. Phys. 57, 2546 (2018)

    MathSciNet  Article  Google Scholar 

  14. Nagata, K., Nakamura, T., Geurdes, H., Batle, J., Abdalla, S., Farouk, A., Diep, D.N.: Creating very true quantum algorithms for quantum energy based computing. Int. J. Theor. Phys. 57, 973 (2018)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

We thank Prof. Do Ngoc Diep, Prof. Germano Resconi, and the reviewers of the paper for valuable comments.

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Correspondence to Koji Nagata.

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Nagata, K., Geurdes, H., Patro, S.K. et al. Generalization of the Bernstein–Vazirani algorithm beyond qubit systems. Quantum Stud.: Math. Found. 7, 17–21 (2020). https://doi.org/10.1007/s40509-019-00196-4

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  • DOI: https://doi.org/10.1007/s40509-019-00196-4

Keywords

  • Quantum algorithms
  • Quantum computation
  • Quantum information theory