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IEEE Conference on Decision and Control (CDC)

Quantum Risk-Sensitive Control

Cassius D'Helon, Andrew C. Doherty, M. R. James, Stuart David Wilson

Published Jan 1, 2006• 25 citations
Publisher

Abstract

The purpose of this paper is to describe some recent results concerning optimal feedback control of quantum systems using risk-sensitive performance criteria. We employ quantum stochastic models to describe an important class of quantum systems and define a risk-sensitive criterion for these models. A suitable information state is introduced to solve the optimal control problem using dynamic programming. The results are illustrated by examples

DOI
10.1109/cdc.2006.377440

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The purpose of this paper is to describe some recent results concerning optimal feedback control of quantum systems using risk-sensitive performance criteria. We employ quantum stochastic models to describe an important class of quantum systems and define a risk-sensitive criterion for these models. A suitable information state is introduced to solve the optimal control problem using dynamic programming. The results are illustrated by examples
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The importance of feedback control is being increasingly appreciated in quantum physics and applications. This paper describes the use of optimal control methods in the design of quantum feedback control systems, and in particular the paper formulates and solves a risk-sensitive optimal control problem. The resulting risk-sensitive optimal control is given in terms of an unnormalized conditional state, whose dynamics include the cost function used to specify the performance objective. The risk-sensitive conditional dynamic equation describes the evolution of our knowledge of the quantum system tempered by our purpose for the controlled quantum system. Robustness properties of risk-sensitive controllers are discussed and an example is provided.
Quantum feedback control is the control of the dynamics of a quantum system by feeding back (in real time) the results of monitoring that system. For systems with linear dynamics, the control problem is amenable to exact analysis. In these cases, the quantum system is equivalent to a stochastic system of classical phase-space variables with linear drift and constant diffusion, and with a measured current (e.g. a homodyne photocurrent) linear in the system variables. However, the classical evolution is constrained in order to represent valid quantum evolution. We quantify this in terms of a linear matrix inequality (LMI) relating the drift and diffusion (a sort of zero temperature fluctuation-dissipation theorem), and another LMI for the covariance matrix of the possible conditioned states (i.e. under all possible monitoring schemes consistent with the master equation). For manipulable systems (i.e. where the experimenter has arbitrary control over the parameters in a Hamiltonian linear in the system variables) the covariance of the conditioned state is all that is needed to calculate the effectiveness of the feedback. In this case the double optimization problem reduces to a semidefinite program, which can be solved efficiently in general. We illustrate this with an example drawn from quantum optics.
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For quantum systems with linear dynamics in phase space much of classical feedback control theory applies. However, there are some questions that are sensible only for the quantum case: Given a fixed interaction between the system and the environment what is the optimal measurement on the environment for a particular control problem? We show that for a broad class of optimal (state-based) control problems (the stationary linear-quadratic-Gaussian class), this question is a semidefinite program. Moreover, the answer also applies to Markovian (current-based) feedback.
The aim of this note is to study the quantum robust optimal control problem for a class of linear quantum systems with uncertainties. It is shown that such problem can be converted into a mixed linear quadratic Gaussian andH∞H^{\infty }H∞quantum control problem. A physical model described by complex quantum stochastic differential equations (CQSDEs) is presented for the quantum system with uncertainties. The quantum system described by CQSDEs is called a complex quantum system in terms of annihilation and creation operators (Maalouf and Petersen, 2011). For an uncertain quantum system, we build a connection with a scaled linear system without uncertainty. The desired quantum robust optimal control results are derived based on the connection. We provide one numerical procedure for quantum robust optimal controller synthesis and then provide an example. The resulting controller presents robustness and optimal performance.