  We examine the number T of queries that a quantum network requires to compute
several Boolean functions on {0,1}^N in the black-box model. We show that, in
the black-box model, the exponential quantum speed-up obtained for partial
functions (i.e. problems involving a promise on the input) by Deutsch and Jozsa
and by Simon cannot be obtained for any total function: if a quantum algorithm
computes some total Boolean function f with bounded-error using T black-box
queries then there is a classical deterministic algorithm that computes f
exactly with O(T^6) 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.
