We consider the problem of search of an unstructured list for a marked element x, when one is given advice as to where x might be located, in the form of a probability distribution. The goal is to minimise the expected number of queries to the list made to find x, with respect to this distribution. We present a quantum algorithm which solves this problem using an optimal number of queries, up to a constant factor. For some distributions on the input, such as certain power law distributions, the algorithm can achieve exponential speed-ups over the best possible classical algorithm. We also give an efficient quantum algorithm for a variant of this task where the distribution is not known in advance, but must be queried at an additional cost. The algorithms are based on the use of Grover’s quantum search algorithm and amplitude amplification as subroutines.


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  1. 1.
    Ambainis, A., de Wolf, R.: Average-case quantum query complexity. J. Phys. A: Math. Gen. 34, 6741–6754 (2001), quant-ph/9904079MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Beals, R., Buhrman, H., Cleve, R., Mosca, M., de Wolf, R.: Quantum lower bounds by polynomials. J. ACM 48(4), 778–797 (2001), quant-ph/9802049MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Boyer, M., Brassard, G., Høyer, P., Tapp, A.: Tight bounds on quantum searching. Fortschr. Phys. 46(4-5), 493–505 (1998), quant-ph/9605034CrossRefGoogle Scholar
  4. 4.
    Brassard, G., Høyer, P., Mosca, M., Tapp, A.: Quantum amplitude amplification and estimation. In: Quantum Computation and Quantum Information: A Millennium Volume, pp. 53–74 (2002), quant-ph/0005055Google Scholar
  5. 5.
    Buhrman, H., de Wolf, R.: Complexity measures and decision tree complexity: a survey. Theoretical Computer Science 288, 21–43 (2002)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Grover, L.: Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 79(2), 325–328 (1997), quant-ph/9706033CrossRefGoogle Scholar
  7. 7.
    Grover, L., Rudolph, T.: Creating superpositions that correspond to efficiently integrable probability distributions (2002), quant-ph/0208112Google Scholar
  8. 8.
    Høyer, P.: Arbitrary phases in quantum amplitude amplification. Phys. Rev. A 62, 052304 (2000), quant-ph/0006031CrossRefGoogle Scholar
  9. 9.
    Motwani, R., Raghavan, P.: Randomized algorithms. Cambridge University Press, Cambridge (1995)CrossRefMATHGoogle Scholar
  10. 10.
    Nielsen, M.A., Chuang, I.L.: Quantum computation and quantum information. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  11. 11.
    Press, W.H.: Strong profiling is not mathematically optimal for discovering rare malfeasors. Proceedings of the National Academy of Sciences 106(6), 1716–1719 (2009)CrossRefGoogle Scholar
  12. 12.
    Zalka, C.: Grover’s quantum searching algorithm is optimal. Phys. Rev. A. 60(4), 2746–2751 (1999), quant-ph/9711070CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Ashley Montanaro
    • 1
  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeUK

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