Quantum Queries on Permutations

  • Taisia Mischenko-Slatenkova
  • Alina Vasilieva
  • Ilja Kucevalovs
  • Rūsiņš FreivaldsEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9118)


K. Iwama and R. Freivalds considered query algorithms where the black box contains a permutation. Since then several authors have compared quantum and deterministic query algorithms for permutations. It turns out that the case of \(n\)-permutations where \(n\) is an odd number is difficult. There was no example of a permutation problem where quantization can save half of the queries for \((2m+1)\)-permutations if \(m\ge 2\). Even for \((2m)\)-permutations with \(m\ge 2\), the best proved advantage of quantum query algorithms is the result by Iwama/Freivalds where the quantum query complexity is \(m\) but the deterministic query complexity is \((2m-1)\). We present a group of \(5\)-permutations such that the deterministic query complexity is 4 and the quantum query complexity is 2.


  1. 1.
    Ablayev, F.M., Freivalds, R.: Why sometimes probabilistic algorithms can be more effective. In: Gruska, J., Rovan, B., Wiedermann, J. (eds.) MPCS 1986. LNCS, vol. 233, pp. 1–14. Springer, Heidelberg (1986)CrossRefGoogle Scholar
  2. 2.
    Ambainis, A.: Quantum lower bounds by quantum arguments. J. Comput. Syst. Sci. 64(4), 750–767 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Ambainis, A.: Polynomial degree vs. quantum query complexity. In: Proceedings of FOCS 1998, pp. 230–240 (1998)Google Scholar
  4. 4.
    Ambainis, A., Freivalds, R.: 1-way quantum finite automata: strengths, weaknesses and generalizations. In: Proceedings of FOCS 1998, pp. 332– 341. Also quant-ph/9802062
  5. 5.
    Ambainis, A., de Wolf, R.: Average-case quantum query complexity. In: Reichel, H., Tison, S. (eds.) STACS 2000. LNCS, vol. 1770, pp. 133–144. Springer, Heidelberg (2000) CrossRefGoogle Scholar
  6. 6.
    Buhrman, H., de Wolf, R.: Complexity measures and decision tree complexity: a survey. Theoret. Comput. Sci. 288(1), 21–43 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Beals, R., Buhrman, H., Cleve, R., Mosca, M., de Wolf, R.: Quantum lower bounds by polynomials. J. ACM 48(4), 778–797 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Buhrman, H., Cleve, R., de Wolf, R., Zalka, C.: Bounds for small-error and zero-error quantum algorithms. In: Proceedings of FOCS 1999, pp. 358–368 (1999)Google Scholar
  9. 9.
    Cleve, R., Ekert, A., Macchiavello, C., Mosca, M.: Quantum algorithms revisited. Proc. R. Soc. Lond. A 454, 339–354 (1998)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Deutsch, D., Jozsa, R.: Rapid solutions of problems by quantum computation. Proc. R. Soc. Lond. A 439, 553 (1992)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Freivalds, R.: Languages recognizable by quantum finite automata. In: Farré, J., Litovsky, I., Schmitz, S. (eds.) CIAA 2005. LNCS, vol. 3845, pp. 1–14. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  12. 12.
    Freivalds, R., Iwama, K.: Quantum queries on permutations with a promise. In: Maneth, S. (ed.) CIAA 2009. LNCS, vol. 5642, pp. 208–216. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  13. 13.
    Simon, I.: String matching algorithms and automata. In: Bundy, A. (ed.) CADE 1994. LNCS, vol. 814, pp. 386–395. Springer, Heidelberg (1994) CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Taisia Mischenko-Slatenkova
    • 1
  • Alina Vasilieva
    • 2
  • Ilja Kucevalovs
    • 2
  • Rūsiņš Freivalds
    • 1
    • 2
    Email author
  1. 1.Institute of Mathematics and Computer ScienceUniversity of LatviaRigaLatvia
  2. 2.Faculty of ComputingUniversity of LatviaRigaLatvia

Personalised recommendations