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

Andris Ambainis, Kazuo Iwama, Masaki Nakanishi, Harumichi Nishimura,
Rudy Raymond, Seiichiro Tani, Shigeru Yamashita

Arxiv ID: 0908.2468•Last updated: 6/24/2020

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).

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