Quantum Boolean Summation with Repetitions in the Worst-Average Setting

We study the quantum summation QS algorithm of Brassard, Hoyer, Mosca and Tapp, 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 $L_q$ norm, $q \in [1,\infty]$. We prove that the QS algorithm with $M$ quantum queries, $M<N$, has the worst-average error bounds of the form $\Theta(\ln M/M)$ for $q=1$, $\Theta(M^{-1/q})$ for $q\in (1,\infty)$, and is equal to 1 for $q=\infty$. 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 $q \in [1,\infty)$. Since $\Omega(M^{-1})$ 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.