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Hamilton-jacobi-bellman equations for quantum optimal control

John Edward Gough, V.A. Belavkin, Oleg Georgievich Smolyanov

Published Jan 1, 2006• 2 citations
Publisher

Abstract

One develops a Hamilton-Jacobi-Bellman equation and a stochastic Hamilton-Jacobi-Bellman equation for quantum optimal feedback control. In this paper we do not discuss in details how to find the optimal control functions; we would like to notice only that the corresponding generalizations of the classical Pontryagin maximum principle can be obtained and used

DOI
10.1109/dd.2006.348159

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One develops a Hamilton-Jacobi-Bellman equation and a stochastic Hamilton-Jacobi-Bellman equation for quantum optimal feedback control. In this paper we do not discuss in details how to find the optimal control functions; we would like to notice only that the corresponding generalizations of the classical Pontryagin maximum principle can be obtained and used
We exploit the separation of the filtering and control aspects of quantum feedback control to consider the optimal control as a classical stochastic problem on the space of quantum states. We derive the corresponding Hamilton-Jacobi-Bellman equations using the elementary arguments of classical control theory and show that this is equivalent, in the Stratonovich calculus, to a stochastic Hamilton-Pontryagin setup. We show that, for cost functionals that are linear in the state, the theory yields the traditional Bellman equations treated so far in quantum feedback.
Interaction of open quantum systems with fundamental noncommutative quantum noises can be described by quantum stochastic differential equations (QSDE). These equations have a key role in quantum network analysis and design, especially for quantum information processing. Hence, in this paper, we derive a Hamilton-Jacobi-Bellman equation for quantum stochastic differential equations. The Bellman optimality principle is developed for open quantum systems. The cost functional of quantum observable to be minimized is considered to be general noncommutative polynomial of quantum operator. Since the method directly deals with QSDE, then it is a useful tool for optimal control of quantum optical networks. In addition, we will exhibit some electro-optical and all-optical feedback control schematics for implementation of quantum control based on QSDEs.
When applied quantization to classical control system, quantum mechanics opens completely new perspectives; by exploiting quantum mechanical features, such as, time evolution of quantum state and costate (generalized momenta) of the Hamiltonian of the system, one can formulate and solve the optimal control problems of the quantum mechanical system more efficiently than the classical system. The physical perspective of quantization of the optimal feedback system is outlined by exploiting the formulation of the optimization problem of the system within the context of Hamilton-Jacobi theory and Pontryagin maximum/minimum principle in abstract Hubert space. One of the most striking example is the optimal control problem of synthesizing the feedback law of quantum diffusion process. The optimal diffusion process is shown to satisfy the Schrödinger wave equation of energy with the feedback law as potential function.
Non‐relativistic quantum systems are analyzed theoretically or by numerical approaches using the Schrödinger equation. Compared to the options available to treat classical mechanical systems this is limited, both in methods and in scope. However, based on Nelson's stochastic mechanics, the mathematical structure of quantum mechanics has in some aspects been developed into a form analogous to classical analytical mechanics. We show here that finding the Nash equilibrium for a stochastic optimal control problem, which is the quantum equivalent to Hamilton's principle of least action, allows to derive two things: i) the Schrödinger equation as the Hamilton‐Jacobi‐Bellman equation of this optimal control problem and ii) a set of quantum dynamical equations which are the generalization of Hamilton's equations of motion to the quantum world. We derive their general form for the non‐stationary and the stationary case. For the harmonic oscillator, the stationary equations lead to the coherent states, and we establish a numerical procedure to solve for the ground state properties without using the Schrödinger equation. image
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 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 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.