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

## TL;DR

This paper investigates the quantum and randomized query complexities of Boolean functions with a fixed number of inputs where the function outputs 1, revealing a super-linear gap in quantum complexity for certain onset sizes.

## Contribution

It establishes the existence of a super-linear gap between average-case and worst-case quantum query complexities for specific onset sizes, and characterizes complexities for various M ranges.

## Key findings

- Super-linear gap between average-case and worst-case quantum complexities for certain M.
- No super-linear gap between average-case and worst-case randomized complexities.
- Quantum complexity is Theta(sqrt{N}) for polynomial M.

## Abstract

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 (sqrt{N}). (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).