Quantum Query Complexity of Boolean Functions with Small On-Sets

The main objective of this paper is to show that the quantum query complexity Q(f) of an N-bit Boolean function f is bounded by a function of a simple and natural parameter, i.e., M = vertical bar{x vertical bar f(x) = 1}vertical bar or the size of f's on-set. We prove that: (i) For poly(N) <= M <= 2(Nd) for some constant 0 < d < 1, the upper bound of Q(f) is O(root N log M/ log N). This bound is tight, namely there is Boolean function f such that Q(f) = Omega(root N log M/ log N). (ii) For the same range of M, the (also tight) lower bound of Q(f) is Omega(root N). (iii) The average value of Q(f) is bounded from above and below by Q(f) = O(log M + root N) and Q(f) = Omega(log M/ log N + root N), respectively. The first bound gives a simple way of bounding the quantum query complexity of testing some graph properties. In particular, it is proved that the quantum query complexity of planarity testing for a graph with n vertices Theta(N-3/4) where N = n(n-1)/2.