How to best sample a periodic probability distribution, or on the accuracy of Hamiltonian finding strategies

TL;DR

This paper investigates optimal experimental strategies for estimating the period of a quantum system's probability distribution, proposing heuristics that scale efficiently and extend to systems with relaxation effects.

Contribution

It introduces heuristic experiment design methods for periodic quantum distributions that match the efficiency of fully optimized strategies and generalizes to finite relaxation times.

Findings

Heuristic strategies achieve exponential scaling similar to optimal methods.

Proposed methods improve accuracy in estimating quantum Hamiltonian parameters.

Extensions to systems with finite relaxation times are discussed.

Abstract

Projective measurements of a single two-level quantum mechanical system (a qubit) evolving under a time-independent Hamiltonian produce a probability distribution that is periodic in the evolution time. The period of this distribution is an important parameter in the Hamiltonian. Here, we explore how to design experiments so as to minimize error in the estimation of this parameter. While it has been shown that useful results may be obtained by minimizing the risk incurred by each experiment, such an approach is computationally intractable in general. Here, we motivate and derive heuristic strategies for experiment design that enjoy the same exponential scaling as fully optimized strategies. We then discuss generalizations to the case of finite relaxation times, T_2 < \infty.