Quantum Query Complexity of Almost All Functions with Fixed On-set Size

This paper considers the quantum query complexity of almost all functions in the set ℱ_N,M of N -variable Boolean functions with on-set size M (1≤ M ≤ 2^N/2) , where the on-set size is the number of inputs on which the function is true. The main result is that, for all functions in ℱ_N,M except its polynomially small fraction, the quantum query complexity is Θ(logM/c + logN - loglogM + √(N)) for a constant c > 0 . This is quite different from the quantum query complexity of the hardest function in ℱ_N,M : Θ(√(NlogM/c + logN - loglogM) + √(N)) . In contrast, almost all functions in ℱ_N,M have the same randomized query complexity Θ(N) as the hardest one, up to a constant factor.