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arXiv:1406.5260

Quantum Control Theory

M. R. James

Published Jun 20, 2014• 3 citations
arXivHTMLPDF

Abstract

This paper explains some fundamental ideas of {\em feedback} control of quantum systems through the study of a relatively simple two-level system coupled to optical field channels. The model for this system includes both continuous and impulsive dynamics. Topics covered in this paper include open and closed loop control, impulsive control, optimal control, quantum filtering, quantum feedback networks, and coherent feedback control.

DOI
10.48550/arxiv.1406.5260

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This paper explains some fundamental ideas of {\em feedback} control of quantum systems through the study of a relatively simple two-level system coupled to optical field channels. The model for this system includes both continuous and impulsive dynamics. Topics covered in this paper include open and closed loop control, impulsive control, optimal control, quantum filtering, quantum feedback networks, and coherent feedback control.
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 review paper, we survey the main concepts and some of the recent developments in quantum feedback control. For consistency and clarity, essential ideas and notations in the theory of open quantum systems and quantum stochastic calculus, as well as continuous measurement theory are developed. We give a general description of quantum feedback control, set up a coherent model and compare it to open-loop designs. Objectives which can be achieved by feedback, such as rapid state preparation and purification or entanglement generation are formulated and analyzed, based on the relevant literature. The connection between quantum feedback and quantum chaos is also described and unravelled which, apart from its theoretical curiosity, can shed more light on some of the intrinsic properties of this control paradigm.
This paper provides a tutorial introduction to quantum feedback control. The paper begins with a discussion of various types of quantum control, and a summary of some relevant aspects of quantum mechanics. We then consider quantum filtering and measurement feedback optimal control for a two-level quantum system. Finally we discuss linear quantum systems, and two types of quantum LQG optimal control problems, one involving measurement, the other fully coherent. We conclude with some remarks about future research directions.
Feedback plays an essential role in classical control theory, which has been investigated in depth. However, in quantum control theory, the study of feedback is still in its infancy. The paper continues to explore the fundamental characteristics of quantum feedback control for a typical coherent model based on the results of [3] by highlighting the significant differences between quantum feedback control and classical feedback control due to quantum measurement. The paper compares the control performance of quantum feedback control and quantum open-loop control in the cases with and without environment-induced decoherence. It concludes that in some sense, the quantum feedback control is still superior to the quantum open-loop control despite the complicated tradeoff in the feedback case, and further points out that how the quantum measurement-induced-collapse can play a positive role in the quantum feedback control.