On the quantum complexity of integration of a function with unknown singularity

In this paper we study the quantum complexity of the integration of a function with an unknown singularity. We assume that the function has $r$ continuous derivatives, with the derivative of order $r$ being H\"older continuous with the exponent $\rho$ on the whole integration interval except the one singular point. We show that the $\ve$-complexity of this problem is of order $\ve^{-1/(r+\rho+1)}$. Since the classical deterministic complexity of this problem is $\ve^{-1/(r+\rho)}$, quantum computers give a speed-up for this problem for all values of parameters $r$ and $\rho$.