Abstract
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 tutorial aims at providing an introduction to key concepts of optimal control theory that 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 tutorial 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 tutorial covers various quantum control issues and describes their mathematical formulation suitable for optimal control. The connection between the Pontryagin maximum principle and gradient-based optimization algorithms used for high-dimensional quantum systems is described. The optimal solution of different low-dimensional quantum systems is presented in detail, illustrating how the mathematical tools are applied in a practical way.2 MoreReceived 10 December 2020Revised 21 May 2021DOI:https://doi.org/10.1103/PRXQuantum.2.030203Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasCoherent controlOptimization problemsQuantum controlQuantum Information
Highlights
Introduction to the Pontryagin MaximumPrinciple for Quantum Optimal ControlU
In order to connect this tutorial with the current applications of optimal control to high-dimensional quantum systems, we describe the link between the Pontryagin maximum principle (PMP) and the most current implementation of the gradient-based optimization algorithm
This analogy gives nonexperts an intuition of the tools of optimal control that might seem abstract on first reading
Summary
A general problem is to prepare a given quantum state by means of a specific time-dependent electromagnetic pulse. The duration of the process is T = π/2u0 This solution reveals a key problem in optimal control that corresponds to the existence of a minimum. In this example, arbitrarily fast controls can be achieved by considering larger and larger amplitudes u0 and an optimal trajectory minimizing the transfer time does not exist. The dynamical constraint is quite simple since the dimension of the state space is the same as the number of controls, the dynamics can be exactly integrated, and the set of controls satisfying the constraint π/2 is regular This is not the case for a general nonlinear control system for which (1) the Lagrange multiplier (which is not usually constant but a function of time) is not found, and (2) abnormal.
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