不同条件下量子布尔求和的精确误差界
Sharp Error Bounds on Quantum Boolean Summation in Various Settings
摘要：
我们研究了量子求和（QS）算法的Bullard，Hoyer，Mosca和Tapp，近似的算术平均数的布尔函数定义在N个元素。我们改进了文献[1]中给出的最坏概率情形下的误差界，并给出了平均概率情形下的新的误差界。特别是，在最坏的概率设置，我们证明了QS算法使用$M - 1$查询的错误是$3π/(4M)$概率$8/π^2$，这提高了错误界$πM^{-1} + π^2 M^{-2}$的Escherard等人。我们还提出了边界概率$p\in (1/2, 8/π^2]$，并表明他们是尖锐的大$M$和$NM^{-1}$。在平均概率的设置，我们证明了QS算法有错误的顺序$\min\{M^{-1}, N^{-1/2}\}$，如果$M$是整除4。这个界限是最优的，如最近在[10]中所示。对于不能被4整除的M，如果$M \ll N^{1/2}$，则QS算法远不是最优的，因为其误差与$M^{-1}^$成比例。
英文摘要：
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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