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 a"not sign(W (p) (r) ([0, 1] (d) )) (r a a"e (+) (d) ). Then combining this result with our previous one we determine the optimal bound of n-th minimal query error for anisotropic Holder-Nikolskii class a"not sign(H (a) (r) ([0, 1] (d) )) and Sobolev class B(W (a) (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.