We study approximation of embeddings between finite dimensional Lp spaces in the quantum model of computation. For the quantum query complexity of this problem matching (up to logarithmic factors) upper and lower bounds are obtained. The results show that for certain regions of the parameter domain quantum computation can essentially improve the rate of convergence of classical deterministic or randomized approximation, while there are other regions where the best possible rates coincide for all three settings. These results serve as a crucial building block for analyzing approximation in function spaces in a subsequent paper [11]. 1

We study the quantum summation (QS) algorithm of Brassard, Høyer, Mosca and Tapp, see [1], that approximates the arithmetic mean of a Boolean function defined on N elements. We improve error bounds presented in [1] in the worst-probabilistic setting, and present new error bounds in the average-probabilistic setting. In particular, in the worst-probabilistic setting, we prove that the error of the QS algorithm using M −1 quantum queries is 3 4πM −1 with probability 8 π2, which improves the error bound πM −1 +π 2 M −2 of [1]. We also present error bounds with probabilities p ∈ ( 1 8 π 2], and show that they are sharp for large M and NM −1.

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