1 Quantum Boolean Summation withRepetitions in the Worst - Average Setting ?

We study the quantum summation (QS) algorithm of Brassard, HHyer, Mosca and Tapp, see 1], which approximates the arithmetic mean of a Boolean function deened on N elements. We present sharp error bounds of the QS algorithm in the worst-average setting with the average performance measured in the Lq norm, q 2 1; 1]. We prove that the QS algorithm with M quantum queries, M < N, has the worst-average error bounds of the form (ln M=M) for q = 1, (M ?1=q) for q 2 (1; 1), and is equal to 1 for q = 1. We also discuss the asymptotic constants of these estimates. We improve the error bounds by using the QS algorithm with repetitions. Using the number of repetitions which is independent of M and linearly dependent on q, we get the error bound of order M ?1 for any q 2 1; 1). Since (M ?1) is a lower bound on the worst-average error of any quantum algorithm with M queries, the QS algorithm with repetitions is optimal in the worst-average setting. The quantum summation (QS) algorithm of Brassard, HHyer, Mosca and Tapp computes an approximation to the arithmetic mean of the values of a Boolean function deened on a set of N = 2 n elements. An overview of the QS algorithm and its basic properties is presented in the rst two sections of 4]. In Section 1.2 we remind the reader of the facts concerning the QS algorithm that are needed in this paper.