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IEEE Workshop on Advanced Research and Technology in Industry Applications (WARTIA)

Trajectory tracking for quantum systems based on Lyapunov control method

Wei Yang

Published Jan 1, 2016• 0 citations
Publisher

Abstract

We investigat the trajectory tracking problem of quantum systems governed by Liouville equation.With the help of Lyapunov function and LaSlle invariance principle, impulsive control fields are given, which drive the quantum state to a time-variant target state.A numerical simulation experiment on a two-level quantum system shows the rationality of the obtained theoretical results, and the advantage compared to traditional control method.

DOI
10.2991/wartia-16.2016.68

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We investigat the trajectory tracking problem of quantum systems governed by Liouville equation.With the help of Lyapunov function and LaSlle invariance principle, impulsive control fields are given, which drive the quantum state to a time-variant target state.A numerical simulation experiment on a two-level quantum system shows the rationality of the obtained theoretical results, and the advantage compared to traditional control method.
Quantum Lyapunov control is a feedback control methodology to determine control fields applied to control quantum systems. In this paper, with the help of the invariant principle of impulsive systems, we investigate the trajectory tracking for quantum systems with impulsive control fields by two Lyapunov functions based on the distance of quantum states and the average value of Hermitian operator which can be regarded as an observable of a quantum system, propose new control fields to drive the quantum systems in the form of sufficient conditions. One numerical simulation is presented to illustrate the effectiveness of the proposed control method.
For quantum state trajectory tracking of density matrix in Liouville equation of quantum systems, with the help of concept in quantum system control, one can apply unitary transformation both to controlled system and free-evolutionary target system such as to change the time-variant and non-stationary target system into a stationary state. Therefore, the quantum state trajectory tracking problem becomes a steering one. State steering control law of the system transformed is designed by means of the Lyapunov stability theorem. Finally, numerical simulation experiments are given for a five-level energy quantum system. The comparison analysis of original system’s trajectory tracking with other method illustrates the advantage in control time of the method proposed.
A Lyapunov-based approach of trajectory tracking in Liouville equation for quantum systems is proposed. By selecting density matrix of controlled system as the system state, and defining the error between controlled state and target one as an error state, the paper regards this error as a new controlled state of system. In such a way the quantum state trajectory tracking problem is changed into the state regulation problem by driving the error state to zero. The control law is designed according to the Lyapunov stability theorem, which can be applied to track different kinds of quantum states including pure state, superposition state and mixed-state. Finally, numerical simulation experiments and results analysis are given.
In this paper, the finite time tracking issue for a class of closed quantum systems under the action of impulsive control fields is addressed. The aim is to steer the state of the quantum system to the trajectory of a periodic time-varying target. By using the Lyapunov direct method and considering a finite-time control strategy, the modified impulsive tracking controllers are designed such that the error dynamical system converges to zero in finite time. While finite-time tracking implies optimality in convergence time and has better disturbance rejection. Numerical simulations are given to illustrate the effects of system’s trajectory tracking, and also given to demonstrate the superiority of the control laws which are proposed.
We investigate the Lyapunov control of finite-dimensional quantum systems with impulsive control fields, where the studied quantum systems are governed by the Schrödinger equation. By three different Lyapunov functions and the invariant principle of impulsive systems, we study the convergence of quantum systems with impulsive control fields and propose new results for the mentioned quantum systems in the form of sufficient conditions. Two numerical simulations are presented to illustrate the effectiveness of the proposed control method.
A control method for trajectory tracking in non-Markovian open quantum systems is proposed in this paper. By inducing the error between the controlled state and the target state as the error state, the trajectory tracking problem of the quantum system is divided into two steps, which the regulation control of the error state and the keeping control of the error state. With the help of Lyapunov stability theorem, the tracking control law is designed to enable the non-Markovian open quantum system to track the evolutionary target quantum system over time. The control parameters are adjusted by simulation experiments, and the numerical simulation experiments are carried out. The tracking control of the quantum system when the initial states of the controlled system and the initial states of the target system are superposition states is implemented in simulation experiments. The experimental results demonstrate the feasibility of the proposed control method and the analysis is given.