Quantum Algorithms for Evaluating Min-Max Trees

  • Richard Cleve
  • Dmitry Gavinsky
  • D. L. Yonge-Mallo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5106)

Abstract

We present a bounded-error quantum algorithm for evaluating Min-Max trees with \(N^{\frac{1}{2}+o(1)}\) queries, where N is the size of the tree and where the allowable queries are comparisons of the form [xj < xk]. This is close to tight, since there is a known quantum lower bound of \(\Omega(N^{\frac{1}{2}})\).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ambainis, A.: Quantum search with variable times (arXiv:quant-ph/0609168)Google Scholar
  2. 2.
    Ambainis, A.: A nearly optimal discrete query quantum algorithm for evaluating NAND formulas (arXiv:quant-ph/0704.3628)Google Scholar
  3. 3.
    Barnum, H., Saks, M.: A lower bound on the query complexity of read-once functions. Journal of Computer and System Science 69(2), 244–258 (2004)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bennett, C.H., Bernstein, E., Brassard, G., Vazirani, U.: Strengths and weaknesses of quantum computing. SIAM Journal on Computing 26(5), 1510–1523 (1997)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Boyer, M., Brassard, G., Høyer, P., Tapp, A.: Tight bounds on quantum searching. Fortschritte Der Physik 46(4–5), 493–505 (1998)CrossRefGoogle Scholar
  6. 6.
    Childs, A.M., Cleve, R., Jordan, S.P., Yeung, D.L.: Discrete-query quantum algorithm for NAND trees (arXiv:quant-ph/070 (2160))Google Scholar
  7. 7.
    Childs, A.M., Reichardt, B.W., Špalek, R., Zhang, S.: Every NAND formula on N variables can be evaluated in time \(O(N^{\frac{1}{2} + \epsilon})\) (arXiv:quant-ph/0703015)Google Scholar
  8. 8.
    Dürr, C., Høyer, P.: A quantum algorithm for finding the minimum (arXiv:quant-ph/9607014)Google Scholar
  9. 9.
    Farhi, E., Goldstone, J., Gutmann, S.: A Quantum Algorithm for the Hamiltonian NAND Tree (arXiv:quant-ph/0702144)Google Scholar
  10. 10.
    Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the 28th Annual ACM Symposium on Theory of Computing (STOC 1996), pp. 212–219 (1996)Google Scholar
  11. 11.
    Saks, M., Wigderson, A.: Probabilistic Boolean Decision Trees and the Complexity of Evaluating Game Trees. In: Proceedings of the 27th Annual IEEE Symposium on Foundations of Computer Science (FOCS 1986), pp. 29–38 (1986)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Richard Cleve
    • 1
    • 2
  • Dmitry Gavinsky
    • 1
  • D. L. Yonge-Mallo
    • 1
  1. 1.David R. Cheriton School of Computer Science and Institute for Quantum ComputingUniversity of WaterlooCanada
  2. 2.Perimeter Institute for Theoretical PhysicsCanada

Personalised recommendations