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.