Sharp Error Bounds on Quantum Boolean Summation in Various Settings

TL;DR

This paper improves error bounds for the quantum summation algorithm in both worst- and average-probabilistic settings, demonstrating near-optimal performance and identifying limitations based on query count and problem size.

Contribution

It provides tighter error bounds for the quantum Boolean summation algorithm, including new sharp bounds in various probabilistic settings and analysis of optimality.

Findings

Error bound of 3π/(4M) with probability 8/π^2 in worst-probabilistic setting

Bounds with probabilities p in (1/2, 8/π^2] are sharp for large M and N/M

In average-probabilistic setting, error of order min{M^{-1}, N^{-1/2}} when M divisible by 4

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 $3\pi /(4M)$ with probability $8/\pi^2$, which improves the error bound $\pi M^{-1} + \pi^2 M^{-2}$ of Brassard et al. We also present bounds with probabilities $p\in (1/2, 8/\pi^2]$ 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 $M^{-1}^$.