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International Conference on Information Science, Electronics and Electrical Engineering

Tracking control laws for some quantum systems based on nonlinear dynamic inversion method

Guangwen Li, Qiuling Jia

Published Apr 1, 2014• 0 citations
Publisher

Abstract

Since real-valued equations of quantum systems are equivalent to the Schrödinger equation, they can provide a convenient way for theoretical analysis and control design, and finally they can assist in the implementation of quantum control, this paper aims at design tracking control laws for some quantum systems based on Nonlinear Dynamic Inversion method in the case that quantum systems are modeled by the real-valued equations. Firstly, the real-valued equations are given for two-level and three level quantum systems. Secondly, after the characteristics of the quantum system models are thoroughly analyzed, the tracking controllers are designed for these quantum systems based on Nonlinear Dynamic Inversion method, and the exponential convergence of the control system is proved. Finally, simulation studies are made to show how to achieve state transfer and how the states track their goal states by the designed controllers; the convergence and validity of the control systems are also illustrated.

DOI
10.1109/infoseee.2014.6948067

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Since real-valued equations of quantum systems are equivalent to the Schrödinger equation, they can provide a convenient way for theoretical analysis and control design, and finally they can assist in the implementation of quantum control, this paper aims at design tracking control laws for some quantum systems based on Nonlinear Dynamic Inversion method in the case that quantum systems are modeled by the real-valued equations. Firstly, the real-valued equations are given for two-level and three level quantum systems. Secondly, after the characteristics of the quantum system models are thoroughly analyzed, the tracking controllers are designed for these quantum systems based on Nonlinear Dynamic Inversion method, and the exponential convergence of the control system is proved. Finally, simulation studies are made to show how to achieve state transfer and how the states track their goal states by the designed controllers; the convergence and validity of the control systems are also illustrated.
This study aims to facilitate quantum control design by making use of derived real-valued equations, which are equivalent to the Schrödinger equation. First, a pure-state identification approach is presented to show that a quantum state can be determined by a minimum number of real parameters. Second, the real-valued equations are deduced for both two- and three-level systems based on the results of the state parametrisation. The procedure is also extended to n-level systems. Third, with the assistance of the derived real-valued equations, the state transfer problem is investigated based on Lyapunov analysis. The state transfer convergence is analysed. It is further shown how to achieve arbitrary pure state transfers via the Lyapunov method by designing the control. The real-valued equations provide a convenient way for theoretical analysis and control design of quantum systems, and therefore can assist in the implementation of quantum control. The state transfer convergence is analysed. It is further shown how to achieve arbitrary pure-state transfers via the Lyapunov method by designing the control.
This paper aims at finding the real-valued dynamics that is equivalent to the Schrödinger equation and then implementing quantum control by making use of the well developed classical control theory. Firstly, pure state identification approaches are presented for two-level, three-level and n-level systems, respectively. Secondly, based on the discussions on the pure state identification, real-valued dynamics that are equivalent to the Schrödinger equations are deduced for both two-level and three-level systems. Finally, a control strategy based on Lyapunov approach is proposed by making use of the obtained real-valued dynamics. Different from the existing Lyapunov control based on the Schrödinger equation, the proposed control strategy can achieve state convergence to its goal state without any constraints on the internal Hamiltonian. Simulation results are included to demonstrate the effectiveness of the approach.
Based on model reference adaptive control theory, we design a control law for a given closed quantum system model by means of the Lyapunov stability theorem by choosing a time-varying objective function as the reference model. Meanwhile, the problems of singularities and a very large value of the control fields during the tracking are solved by using an adaptive tracking algorithm. The simulation experiments realize the tracking task from the arbitrary initial states to the time-varying objective function. Under the effect of control law, the state of control system can track from the arbitrary initial state to the state of objective system. Simulation results demonstrate the ability of the control method proposed to adaptively adjust the control laws and real-time track dynamic target system.
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 quantum state transfer and tracking, control of quantum states are effective against decoherence caused by external environment and information processing. In this paper, utilizing model reference adaptive control theory and Lyapunov stability theorem, we derive the adaptive law for the model reference adaptive system. Then we design the Lyapunov control law by double control functions and we investigate the orbit tracking of quantum state for Markovian quantum system. Meanwhile, through counteracting the shift term by simplization, the control law avoids oscillating state trajectory in a two-level Markovian quantum system. Finally, numerical simulations for a two-level open quantum system validate the effectiveness of the method.