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FROM CLASSICAL TO QUANTUM QUADRATIC COST CONTROL

Luigi Accardi, Andreas Boukas

Published Oct 1, 2004• 1 citation
Publisher

Abstract

Controlling the size of the solution of a (deterministic, stochastic or quantum stochastic) differential equation, by minimizing an appropriate cost functional, is very important in classical and quantum engineering. Of particular importance is the case of linear differential equations and quadratic cost functionals, since in that case the control processes can be explicitly calculated. In this paper we review some basic aspects of the classical theory and we present our results in the quantum case, obtained over the past few years.

DOI
10.1142/9789812702449_0004

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Controlling the size of the solution of a (deterministic, stochastic or quantum stochastic) differential equation, by minimizing an appropriate cost functional, is very important in classical and quantum engineering. Of particular importance is the case of linear differential equations and quadratic cost functionals, since in that case the control processes can be explicitly calculated. In this paper we review some basic aspects of the classical theory and we present our results in the quantum case, obtained over the past few years.
The problem of controlling quantum stochastic evolutions arises naturally in several different fields such as quantum chemistry, quantum information theory, quantum engineering, etc. In this paper, we apply the recently discovered closed form of the unitarity conditions for stochastic evolutions driven by the square of white noise [9] to solve this problem in the case of quadratic cost functionals (cf. (5.5) below). The optimal control is explicitly given in terms of the solution of an operator Riccati equation. Under general conditions on the system Hamiltonian part of the stochastic evolution and on the system observable to be controlled, this equation admits solutions with the required properties and they can be explicitly described.
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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.