An optimal quantum algorithm to approximate the mean and its application for approximating the median of a set of points over an arbitrary distance

TL;DR

This paper introduces two quantum algorithms for approximating the mean of a black-box function, with one being optimal, and applies these methods to develop a new quantum algorithm for median approximation over arbitrary distances.

Contribution

The paper presents a novel asymptotically optimal quantum algorithm for mean approximation and extends this approach to median approximation for points with arbitrary distance metrics.

Findings

One quantum algorithm is asymptotically optimal for mean approximation.

The second quantum algorithm is a variation of a prior method.

A new quantum algorithm for median approximation over arbitrary distances is proposed.

Abstract

We describe two quantum algorithms to approximate the mean value of a black-box function. The first algorithm is novel and asymptotically optimal while the second is a variation on an earlier algorithm due to Aharonov. Both algorithms have their own strengths and caveats and may be relevant in different contexts. We then propose a new algorithm for approximating the median of a set of points over an arbitrary distance function.