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Annual Review of Control Robotics and Autonomous Systems

Optimal Quantum Control Theory

M. R. James

Published Jan 12, 2021• 24 citations
Publisher

Abstract

This article explains some fundamental ideas concerning the optimal control of quantum systems through the study of a relatively simple two-level system coupled to optical fields. The model for this system includes both continuous and impulsive dynamics. Topics covered include open- and closed-loop control, impulsive control, open-loop optimal control, quantum filtering, and measurement feedback optimal control.

DOI
10.1146/annurev-control-061520-010444

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This article explains some fundamental ideas concerning the optimal control of quantum systems through the study of a relatively simple two-level system coupled to optical fields. The model for this system includes both continuous and impulsive dynamics. Topics covered include open- and closed-loop control, impulsive control, open-loop optimal control, quantum filtering, and measurement feedback optimal control.
Optimal Control Theory is a powerful mathematical tool, which has known a rapid development since the 1950s, mainly for engineering applications. More recently, it has become a widely used method to improve process performance in quantum technologies by means of highly efficient control of quantum dynamics. This review aims at providing an introduction to key concepts of optimal control theory which is accessible to physicists and engineers working in quantum control or in related fields. The different mathematical results are introduced intuitively, before being rigorously stated. This review describes modern aspects of optimal control theory, with a particular focus on the Pontryagin Maximum Principle, which is the main tool for determining open-loop control laws without experimental feedback. The different steps to solve an optimal control problem are discussed, before moving on to more advanced topics such as the existence of optimal solutions or the definition of the different types of extremals, namely normal, abnormal, and singular. The review covers various quantum control issues and describes their mathematical formulation suitable for optimal control. The optimal solution of different low-dimensional quantum systems is presented in detail, illustrating how the mathematical tools are applied in a practical way.
This study presents a survey on quantum control theory and applications from a control systems perspective. Some of the basic concepts and main developments (including open-loop control and closed-loop control) in quantum control theory are reviewed. In the area of open-loop quantum control, the paper surveys the notion of controllability for quantum systems and presents several control design strategies including optimal control, Lyapunov-based methodologies, variable structure control and quantum incoherent control. In the area of closed-loop quantum control, this study reviews closed-loop learning control and several important issues related to quantum feedback control including quantum filtering, feedback stabilisation, linear-quadratic-Gaussian control and robust quantum control.
This chapter provides fundamental concepts and problems in quantum control theory. It discusses physical mechanisms behind quantum control theory. The chapter introduces modeling and analysis problems and summarizes open-loop and feedback control methods. In principle, Bayesian feedback may outperform the Markovian feedback because more information extracted from the measurement history can be used to determine the control. The fusion of control theory and quantum physics calls for collaborations between physicists and engineers. In contrast to classical control theory, the body of literature on open-loop control is much larger than that of feedback control. In quantum optimal control theory, all algorithms may suffer from the possible existence of false traps, that is, the search may converge to an undesired locally optimal solution that is not globally optimal. Quantum optimal control theory has been proven to be effective for finite-level or simple few-body systems.
In this paper, we analyze classical and quantum physical systems from an optimal control perspective. Specifically, we explore whether their associated dynamics can correspond to an open or closed-loop feedback evolution of a control problem. Firstly, for the classical regime, when it is viewed in terms of the theory of canonical transformations, we find that it can be described by a closed-loop feedback problem. Secondly, for a quantum physical system, if one realizes that the Heisenberg commutation relations themselves can be thought of as constraints in a non-commutative space, then the momentum must be dependent on the position for any generic wave function. This implies the existence of a closed-loop strategy for the quantum case. Thus, closed-loop feedback is a natural phenomenon in the physical world. For the sake of exposition, we give a short review of control theory, and some familiar examples at the classical and quantum levels are analyzed.
Using results from quantum filtering theory and methods from classical control theory, we derive an optimal control strategy for an open two-level system (a qubit in interaction with the electromagnetic field) controlled by a laser. The aim is to optimally choose the laser's amplitude and phase in order to drive the system into a desired state. The Bellman equations are obtained for the case of diffusive and counting measurements for vacuum field states. A full exact solution of the optimal control problem is given for a system with simpler, linear, dynamics. These linear dynamics can be obtained physically by considering a two-level atom in a strongly driven, heavily damped, optical cavity.