Average/Worst-Case Gap of Quantum Query Complexities by On-Set Size

This paper considers the query complexity of the functions in the family F_N,M of N-variable Boolean functions with onset size M, i.e., the number of inputs for which the function value is 1, where 1<= M <= 2^N/2 is assumed without loss of generality because of the symmetry of function values, 0 and 1. Our main results are as follows: (1) There is a super-linear gap between the average-case and worst-case quantum query complexities over F_N,M for a certain range of M. (2) There is no super-linear gap between the average-case and worst-case randomized query complexities over F_N,M for every M. (3) For every M bounded by a polynomial in N, any function in F_N,M has quantum query complexity Theta (sqrtN). (4) For every M=O(2^cN) with an arbitrary large constant c<1, any function in F_N,M has randomized query complexity Omega (N).