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Sharp Error Bounds on Quantum Boolean Summation in Various Settings

Marek Kwas·Henryk Wozniakowski
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Abstract

We study the quantum summation (QS) algorithm of Brassard, Hoyer, Mosca and Tapp, 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 $M - 1$ queries is $3π/(4M)$ with probability $8/π^2$, which improves the error bound $πM^{-1} + π^2 M^{-2}$ of Brassard et al. We also present bounds with probabilities $p\in (1/2, 8/π^2]$ and show they are sharp for large $M$ and $NM^{-1}$. In the average-probabilistic setting, we prove that the QS algorithm has error of order $\min\{M^{-1}, N^{-1/2}\}$ if $M$ is divisible by 4. This bound is optimal, as recently shown in [10]. For M not divisible by 4, the QS algorithm is far from being optimal if $M \ll N^{1/2}$ since its error is proportional to $M^{-1}^$.

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Marek Kwas, Henryk Wozniakowski. 2003-03-10. Sharp Error Bounds on Quantum Boolean Summation in Various Settings. https://arxiv.org/abs/quant-ph/0303049

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