Sobolev Approximation in the Quantum Computation Model

Abstract

Using a new and elegant reduction approach we derive a lower bound of quantum complexity for the approximation of imbeddings from anisotropic Sobolev classes B(Wrp([0,1]d)) to anisotropic Sobolev space Wsp([0,1]d) for all 1 ? p, q ? ?, When p ? q we show this bound is optimal by deriving the matching upper bound. In this case the quantum algorithms are not significantly better than the classical deterministic or randomized algorithms.