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

Abstract

This paper considers the quantum query complexity of almost all functions in the set ${\mathcal{F}}_{N,M}$ of ${N}$-variable Boolean functions with on-set size ${M (1\le M \le 2^{N}/2)}$, 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 ${\mathcal{F}}_{N,M}$ except its polynomially small fraction, the quantum query complexity is ${ \Theta\left(\frac{\log{M}}{c + \log{N} - \log\log{M}} + \sqrt{N} \right)}$ for a constant ${c > 0}$. This is quite different from the quantum query complexity of the hardest function in ${\mathcal{F}}_{N,M}$: ${\Theta\left(\sqrt{N\frac{\log{M}}{c + \log{N} - \log\log{M}}} + \sqrt{N} \right)}$. In contrast, almost all functions in ${\mathcal{F}}_{N,M}$ have the same randomized query complexity ${\Theta(N)}$ as the hardest one, up to a constant factor.