Convergence rate of quantum algorithm for multivariate approximation

We estimate the convergence rate of quantum algorithm for approximation from some smooth functions in the L_q([0, 1]^d) norm for 1 ≤ q ≤ ∞. It turns out that for the Sobolev class B(W_p^r([0, 1]^d)) (r ∈ ℕ^d), when p <; q, the quantum algorithms can bring speedup over classical deterministic and randomized algorithms.