Abstract
A basic problem of approximation theory, the approximation of functions from the Sobolev space W p r ([0,1] d ) in the norm of L q ([0,1] d ), is considered from the point of view of quantum computation. We determine the quantum query complexity of this problem (up to logarithmic factors). It turns out that in certain regions of the domain of parameters p, q, r, d quantum computation can reach a speedup of roughly squaring the rate of convergence of classical deterministic or randomized approximation methods. There are other regions where the best possible rates coincide for all three settings.
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