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International series of numerical mathematics

Nonlinear Feedback Solutions for a Class of Quantum Control Problems

Kazufumi Ito, Karl K. Kunisch, Qin Zhang

Published Jan 1, 2009• 1 citation
Publisher

Abstract

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.

DOI
10.1007/978-3-7643-8923-9_9

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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.
The paper proposes a gradient method for feedback control and stabilization of quantum systems using Schröodinger’s and Lindblad’s descriptions. The eigenstates of the quantum system are defined by the spin model. First, a gradient-based control law is computed using Schröodinger’s description. Next, an estimate of state of the quantum system is obtained using Lindblad’s differential equation. In the latter case, by applying Lyapunov’s stability theory and LaSalle’s invariance principle one can compute a gradient control law which assures that the quantum system’s state will track the desirable state within acceptable accuracy levels. The performance of the control loop is studied through simulation experiments for the case of a two-qubit quantum system.
We present detailed analysis of the convergence properties and effectiveness of Lyapunov control design for bilinear Hamiltonian quantum systems based on the application of LaSalle's invariance principle and stability analysis from dynamical systems and control theory. For a certain class of Hamiltonians, strong convergence results can be obtained for both pure and mixed state systems. The control Hamiltonians for realistic physical systems, however, generally do not fall in this class. It is shown that the effectiveness of Lyapunov control design in this case is significantly diminished.
In this paper, we investigate a Lyapunov trajectory tracking design method that incorporates a Schrödinger equation with a dipole subterm and polarizability. Our findings suggest that the proposed control law can overcome the limitations of certain existing control laws that do not converge. By integrating a quadratic performance index, we introduce an optimal control law, which we subsequently analyze for stability and optimality. We also simulate the spin-1/2 particle system to illustrate our results. These findings are further validated through numerical illustrations involving a 3D, 5D system, and a spin-1/2 particle system.
This thesis is concerned with the control and filtering problems for quantum systems. Two kinds of control problems are considered for a class of quantum systems: optimal control and trajectory tracking control. Filtering of a class of linear quantum systems is also investigated. \n \nFor the optimal control of quantum systems, the system in question evolves on a manifold inR^3 and is modeled as a bilinear control form whose states are represented as coherence vectors. An associated matrix Lie group system with state space S O(3) is introduced in order to facilitate solving the given problem. The controllability as well as the reachable set of the system is analyzed in detail. The maximum principle is applied to the optimal control for systems evolving on the Lie group of special orthogonal matrices of dimension 3, with cost that is quadratic in the control input. Controlling the evolutions of coherence vector is equivalent to controlling the states of an associated matrix Lie group system. Thus, the optimal control problem for single spin-1/2 systems can be transformed to steer state of the matrix Lie group system from the identity matrix to a final matrix corresponding to a target state of single spin-1/2 systems in an optimal fashion. The obtained result is employed to perform a reversible logic quantum operation NOT on single spin-1/2 systems. Explicit expressions for the optimal control are given which are linked to the initial state of the system. The main contribution of the result is to extend the existing result to a larger class of systems, generalizing existing wave-function-based results to single spin-1/2 systems whose states are represented by density matrices such that it allows the inclusion of mixed states. \n \nFor the tracking control problem of closed single spin-1/2 quantum ensembles, based on the Liouville-von Neumann equation, mathematical model of dynamics for single spin-1/2 quantum ensembles are formulated as a bilinear form with states evolving on Bloch sphere. For a typical setting in NMR, one in general would not cancel the applied electromagnetic field along a specified direction, thus leading to the dynamics in a plane perpendicular to that direction on the Bloch sphere. One natural question arises that how can one design a feedback controller such that given an initial condition for quantum ensembles, the system evolutions can be steered to the pre-determined trajectory. An associated matrix Lie group differential equation describing transfers from the initial state being an identity matrix is naturally introduced. The tracking problem for the system of interest is transformed to a stabilization problem such that the control techniques are able to be used. A distance function as well as a distance vector defined on the Lie group SO(3) are introduced, which can be used to construct a Lyapunov function to deal with the tracking problem. In order to deal with the tracking problem, the Lyapunov-based feedback design method is employed to obtain the feedback control such that the dynamics of system can converge to the reference trajectory. \n \nFor filtering of quantum systems, the filter problem of a class of linear quantum stochastic systems with quantum noise is investigated. The quantum system model is first analyzed, then the necessary conditions of nondemolition are obtained by applying the observation principle. The relations between the input and output noise are necessary for the nondemolition measurement of the quantum output channels. Given prescribed quantum system, the filter of the system is derived using those necessary conditions.
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