Optimal Experiment Design for Quantum State and Process Tomography and Hamiltonian Parameter Estimation

TL;DR

This paper develops convex optimization methods for optimal experiment design in quantum state, process, and Hamiltonian parameter estimation, enabling efficient and accurate quantum system identification.

Contribution

It extends maximum likelihood estimation to quantum process tomography and Hamiltonian estimation, and formulates optimal experiment design as a convex optimization problem.

Findings

MLE applies to quantum process and Hamiltonian estimation

Optimal experiment design can be solved via convex optimization

Software tools are provided for implementation

Abstract

A number of problems in quantum state and system identification are addressed. Specifically, it is shown that the maximum likelihood estimation (MLE) approach, already known to apply to quantum state tomography, is also applicable to quantum process tomography (estimating the Kraus operator sum representation (OSR)), Hamiltonian parameter estimation, and the related problems of state and process (OSR) distribution estimation. Except for Hamiltonian parameter estimation, the other MLE problems are formally of the same type of convex optimization problem and therefore can be solved very efficiently to within any desired accuracy. Associated with each of these estimation problems, and the focus of the paper, is an optimal experiment design (OED) problem invoked by the Cramer-Rao Inequality: find the number of experiments to be performed in a particular system configuration to maximize estimation accuracy; a configuration being any number of combinations of sample times, hardware settings, prepared initial states, etc. We show that in all of the estimation problems, including Hamiltonian parameter estimation, the optimal experiment design can be obtained by solving a convex optimization problem. Software to solve the MLE and OED convex optimization problems is available upon request from the first author.