On the quantum complexity of approximating median of continuous distribution

Abstract

We consider approximating of the median of absolutely continuous distribution given by a probability density function $f$. We assume that $f$ has $r$ continuous derivatives, with derivative of order $r$ being Hölder continuous with the exponent $ρ$. We study the $ε$-complexity of this problem in the quantum setting. We show that the $ε$-complexity up to logarithmic factor is of order $ε^{-1/((r+ρ+1))}$.