Average-Case Quantum Query Complexity

  • Andris Ambainis
  • Ronald de Wolf
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1770)


We compare classical and quantum query complexities of total Boolean functions. It is known that for worst-case complexity, the gap between quantum and classical can be at most polynomial [3]. We show that for average-case complexity under the uniform distribution, quantum algorithms can be exponentially faster than classical algorithms. Under non-uniform distributions the gap can even be super-exponential. We also prove some general bounds for average-case complexity and show that the average-case quantum complexity of MAJORITY under the uniform distribution is nearly quadratically better than the classical complexity.


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  1. 1.
    N. Alon and J. H. Spencer. The Probabilistic Method. Wiley-Interscience, 1992.Google Scholar
  2. 2.
    L. Alonso, E. M. Reingold, and R. Schott. The average-case complexity of determining the majority. SIAM Journal on Computing, 26(1):1–14, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    R. Beals, H. Buhrman, R. Cleve, M. Mosca, and R. de Wolf. Quantum lower bounds by polynomials. In Proceedings of 39th FOCS, pages 352–361, 1998.
  4. 4.
    C. H. Bennett, E. Bernstein, G. Brassard, and U. Vazirani. Strengths and weaknesses of quantum computing. SIAM Journal on Computing, 26(5):1510–1523, 1997. quant-ph/9701001.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    M. Boyer, G. Brassard, P. Høyer, and A. Tapp. Tight bounds on quantum searching. Fortschritte der Physik, 46(4–5):493–505, 1998. Earlier version in Physcomp’96. quant-ph/9605034.CrossRefGoogle Scholar
  6. 6.
    G. Brassard, P. Høyer, M. Mosca, and A. Tapp. Quantum amplitude amplification and estimation. Forthcoming.Google Scholar
  7. 7.
    G. Brassard, P. Høyer, and A. Tapp. Quantum algorithm for the collision problem. ACM SIGACT News (Cryptology Column), 28:14–19, 1997. quant-ph/9705002.CrossRefGoogle Scholar
  8. 8.
    G. Brassard, P. Høyer, and A. Tapp. Quantum counting. In Proceedings of 25th ICALP, volume 1443 of Lecture Notes in Computer Science, pages 820–831. Springer, 1998. quant-ph/9805082.Google Scholar
  9. 9.
    D. Deutsch and R. Jozsa. Rapid solution of problems by quantum computation. In Proceedings of the Royal Society of London, volume A439, pages 553–558, 1992.MathSciNetGoogle Scholar
  10. 10.
    E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser. A limit on the speed of quantum computation in determining parity. quant-ph/9802045, 16 Feb 1998.Google Scholar
  11. 11.
    L. K. Grover. A fast quantum mechanical algorithm for database search. In Proceedings of 28th STOC, pages 212–219, 1996. quant-ph/9605043.Google Scholar
  12. 12.
    E. Hemaspaandra, L. A. Hemaspaandra, and M. Zimand. Almost-everywhere superiority for quantum polynomial time. quant-ph/9910033, 8 Oct 1999.Google Scholar
  13. 13.
    L. A. Levin. Average case complete problems. SIAM Journal on Computing, 15(1):285–286, 1986. Earlier version in STOC’84.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    M. Mosca. Quantum searching, counting and amplitude amplification by eigenvector analysis. In MFCS’98 workshop on Randomized Algorithms, 1998.Google Scholar
  15. 15.
    A. Nayak and F. Wu. The quantum query complexity of approximating the median and related statistics. In Proceedings of 31th STOC, pages 384–393, 1999. quant-ph/9804066.Google Scholar
  16. 16.
    N. Nisan. CREW PRAMs and decision trees. SIAM Journal on Computing, 20(6):999–1007, 1991. Earlier version in STOC’89.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    P. W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing, 26(5):1484–1509, 1997. Earlier version in FOCS’94. quant-ph/9508027.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    D. Simon. On the power of quantum computation. SIAM Journal on Computing, 26(5):1474–1483, 1997. Earlier version in FOCS’94.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    J. S. Vitter and Ph. Flajolet. Average-case analysis of algorithms and data structures. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science. Vol-ume A: Algorithms and Complexity, pages 431–524. MIT Press, Cambridge, MA, 1990.Google Scholar
  20. 20.
    Ch. Zalka. Grover’s quantum searching algorithm is optimal. Physical Review A, 60:2746–2751, 1999. quant-ph/9711070.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Andris Ambainis
    • 1
  • Ronald de Wolf
    • 2
    • 3
  1. 1.Computer Science DepartmentUniversity of CaliforniaBerkeley
  2. 2.CWIAmsterdamThe Netherlands
  3. 3.ILLCUniversity of AmsterdamAmsterdam

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