Quantum lower bounds by polynomials

  • Authors:

  • Robert Beals;Harry Buhrman;Richard Cleve;Michele Mosca;Ronald de Wolf

  • Affiliations:

  • University of Arizona, Tucson, Arizona;CWI and University of Amsterdam, Amsterdam, The Netherlands;University of Calgary, Calgary, Alberta, Canada;University of Waterloo, Waterloo, Canada;CWI and University of Amsterdam, Amsterdam, The Netherlands

  • Venue:
  • Journal of the ACM (JACM)
  • Year:
  • 2001

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Abstract

We examine the number of queries to input variables that a quantum algorithm requires to compute Boolean functions on {0,1}N in the black-box model. We show that the exponential quantum speed-up obtained for partial functions (i.e., problems involving a promise on the input) by Deutsch and Jozsa, Simon, and Shor cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with small error probability using T black-box queries, then there is a classical deterministic algorithm that computes f exactly with O(Ts6) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the so-called polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized and deterministic decision tree complexity.