Lower Bound for Quantum Integration Error on Some 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 classes B(W_p ^r ([0, 1]^d)), (r isin R₊ ^d). Then combining our previous results we determine the optimal bound of nth minimal query error for anisotropic Holder-Nikolskii class B(H_infin ^r([0, 1]^d)), (r isin R₊ ^d) and Sobolev class B(W_p ^r([0, 1]^d)) (r isin N^d). The results show the quantum algorithms give speed up over classical algorithms.