Fourier Analytic Approach to Phase Estimation

TL;DR

This paper analyzes phase estimation using Fourier transforms, revealing the relationship between the limiting distribution's variance and tail probability, and deriving optimal protocols with minimal tail probability and confidence intervals.

Contribution

It introduces a Fourier analytic framework for phase estimation, identifying optimal protocols based on prolate spheroidal wave functions and analyzing their statistical properties.

Findings

The limiting distribution is the squared Fourier transform of an $L^2$ function supported on [-1,1].

Protocols minimizing asymptotic variance do not minimize tail probability.

Optimal protocols for tail probability are derived using prolate spheroidal wave functions.

Abstract

For a unified analysis on the phase estimation, we focus on the limiting distribution. It is shown that the limiting distribution can be given by the absolute square of the Fourier transform of $L^2$ function whose support belongs to $[-1,1]$. Using this relation, we study the relation between the variance of the limiting distribution and its tail probability. As our result, we prove that the protocol minimizing the asymptotic variance does not minimize the tail probability. Depending on the width of interval, we derive the estimation protocol minimizing the tail probability out of a given interval. Such an optimal protocol is given by a prolate spheroidal wave function which often appears in wavelet or time-limited Fourier analysis. Also, the minimum confidence interval is derived with the framework of interval estimation that assures a given confidence coefficient.