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Complexity for the approximation of Sobolev imbeddings in the quantum computation model

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(W<inf>p</inf><sup>r</sup>([0, 1]<sup>d</sup>)) to anisotropic Sobolev space W<inf>p</inf><sup>s</sup>([0, 1]<sup>d</sup>) for all 1 ≤ p, q ≤ ∞. When p ≥ q this bound is optimal. In this case the quantum algorithms are not significantly better than the classical deterministic or randomized algorithms. When p ≥ q we conjecture that quantum algorithms bring speed-up over the classical deterministic and randomized ones. This conjecture was confirmed in the situation s = 0.