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SIAM Journal on Control and Optimization

Asymptotic Properties of Feedback Solutions for a Class of Quantum Control Problems

Kazufumi Ito, Karl K. Kunisch

Published Jan 1, 2009• 6 citations
Publisher

Abstract

Control of quantum systems described by the Schrödinger equation is considered. Feedback control laws are developed for orbit tracking via controlled Hamiltonians, and their asymptotic properties are analyzed. Numerical integration via time splitting is also investigated and used to demonstrate the feasibility of the proposed feedback laws.

DOI
10.1137/080720784

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Control of quantum systems described by the Schrödinger equation is considered. Feedback control laws are developed for orbit tracking via controlled Hamiltonians, and their asymptotic properties are analyzed. Numerical integration via time splitting is also investigated and used to demonstrate the feasibility of the proposed feedback laws.
Control of quantum systems described by the linear Schrödinger equation are considered. Control inputs enter through coupling operators and result in a bilinear control system. Feedback control laws are developed for orbit tracking. The asymptotic properties of the feedback laws are analyzed by the LaSalle-type invariance principle. Numerical integration via time-splitting is also investigated and used to demonstrate the feasibility of the proposed feedback laws and to compare their performance.
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No quantum measurement can give full information on the state of a quantum system; hence any quantum feedback control problem is necessarily one with partial observations and can generally be converted into a completely observed control problem for an appropriate quantum filter as in classical stochastic control theory. Here we study the properties of controlled quantum filtering equations as classical stochastic differential equations. We then develop methods, using a combination of geometric control and classical probabilistic techniques, for global feedback stabilization of a class of quantum filters around a particular eigenstate of the measurement operator.
Abstract. No quantum measurement can give full information on the state of a quantum system; hence any quantum feedback control problem is necessarily one with partial observations and can generally be converted into a completely observed control problem for an appropriate quantum filter as in classical stochastic control theory. Here we study the properties of controlled quantum filtering equations as classical stochastic differential equations. We then develop methods, using a combination of geometric control and classical probabilistic techniques, for global feedback stabilization of a class of quantum filters around a particular eigenstate of the measurement operator.
This paper surveys some recent results on the feedback control of quantum linear systems and the robustness properties of these systems. Quantum linear systems are a class of systems whose dynamics, which are described by the laws of quantum mechanics, take the specific form of a set of linear quantum stochastic differential equations (QSDEs). These systems can also be described in terms of a Hamiltonian operator H and a coupling operator L, which in the case of quantum linear systems have a specific quadratic and linear form respectively. Such systems commonly arise in the area of quantum optics and related disciplines. Systems whose dynamics can be described or approximated by linear QSDEs include interconnections of optical cavities, beam-splitters, phase-shifters, optical parametric amplifiers, optical squeezers, and cavity quantum electrodynamic systems. An important approach to the feedback control of quantum linear systems involves the use of a controller which itself is a quantum linear system. This approach to quantum feedback control, referred to as coherent quantum feedback control, has the advantage that it does not destroy quantum information, is fast, and has the potential for efficient implementation. The paper discusses recent results concerning the synthesis of coherent quantum controllers such as in coherent quantum H∞ control. Another important issue in the design of quantum feedback controllers is the robustness of the closed loop quantum system to perturbations in the quantum plant dynamics arising from perturbations in the system Hamiltonian H and the system coupling operator L. The paper discusses recent robust stability results for such perturbed quantum systems which take the form of a quantum small gain theorem and a quantum Popov stability criterion. These results are usefu