Optimal integration error on anisotropic classes for restricted Monte Carlo and quantum algorithms

Authors:
Peixin Ye;Xiaofei Hu
Affiliations:
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China;College of Mathematical, Syracuse University, NY 13210, USA
Venue:
Journal of Approximation Theory
Year:
2008

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Abstract

We study restricted Monte Carlo integration for anisotropic Holder-Nikolskii classes. The results show that with clog"2n random bits we have the same optimal order for the nth minimal Monte Carlo integration error as with arbitrary random numbers. We also study the computation of integration on anisotropic Sobolev classes in the quantum setting and present the optimal bound of nth minimal query error. The results show that the error bound of quantum algorithms is much smaller than that of deterministic and randomized algorithms.