ar X iv : q ua nt - p h / 03 11 03 6 v 1 6 N ov 2 00 3 1 Quantum Boolean Summation with Repetitions in the Worst - Average Setting ⋆
نویسنده
چکیده
We study the quantum summation (QS) algorithm of Brassard, Høyer, Mosca and Tapp, see [1], which approximates the arithmetic mean of a Boolean function defined 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 ∈ [1,∞]. We prove that the QS algorithm with M quantum queries, M < N , has the worst-average error bounds of the form Θ(lnM/M) for q = 1, Θ(M) for q ∈ (1,∞), and is equal to 1 for q = ∞. 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 for any q ∈ [1,∞). Since Ω(M) 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.
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