Lower bound for quantum integration error on anisotropic Sobolev classes

We study the approximation of the integration of multivariate functions in the quantum model of computation. Using a new reduction approach we obtain a lower bound of the n-th minimal query error on anisotropic Sobolev class ℬ(Wpr ([0, 1]d)) (r ∈ ℝ+d). Then combining this result with our previous one we determine the optimal bound of n-th minimal query error for anisotropic Hölder-Nikolskii class ℬ(H∞r ([0, 1]d)) and Sobolev class B(W∞r ([0, 1]d)). The results show that for these two types of classes the quantum algorithms give significant speed up over classical deterministic and randomized algorithms.