Quantum Query Complexity of Boolean Functions with Small On-Sets

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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 = |{x|f(x) = 1}| or the size of f’s on-set. We prove that: (i) For \(poly(N)\le M\le 2^{N^d}\) for some constant 0 < d < 1, the upper bound of Q(f) is \(O(\sqrt{N\log M / \log N})\) . This bound is tight, namely there is a Boolean function f such that \(Q(f) = \Omega(\sqrt{N\log M / \log N})\) . (ii) For the same range of M, the (also tight) lower bound of Q(f) is \(\Omega(\sqrt{N})\) . (iii) The average value of Q(f) is bounded from above and below by \(Q(f) = O(\log M +\sqrt{N})\) and \(Q(f) = \Omega (\log M/\log N+ \sqrt{N})\) , 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 is Θ(N 3/4) where \(N = \frac{n(n-1)}{2}\) .