Entropy lower bounds for quantum decision tree complexity

We prove a general lower bound of quantum decision tree complexity in terms of some entropy notion. We regard decision tree computation as a communication process in which the oracle and the computer exchange several rounds of messages, each round consisting of O(log n) bits. Let E(f) be the Shannon entropy of the random variable f (X), where X is taken uniformly random in f's domain. Our main result is that it takes n (E (f)) queries to compute any total function f. It is interesting to contrast this bound with the Omega (E (f)/log n) bound, which is tight for some partial functions. (C) 2002 Elsevier Science B.V. All rights reserved.