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IEEE Conference on Robotics, Automation and Mechatronics (RAM)

Identification and control of quantum systems

Zhengui Xue, Hai Lin, Tong Heng Lee

Published Jun 1, 2010• 1 citation
Publisher

Abstract

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.

DOI
10.1109/ramech.2010.5513217

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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.
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.
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.
A quantum control theory of pure state in non-ideal quantum systems is proposed in this paper. A non-ideal closed quantum systems means that the controlled quantum system is in two degenerate cases: one is at least two transition frequencies between different energy levels are the same, or/and another is at least two eigenstates of the internal Hamiltonian are not directly coupled. The implicit Lyapunov control method based on the average value of an imaginary mechanical quantity is used to design the control laws. The design procedure of the imaginary mechanical quantity is derived. The relationships among the implicit Lyapunov control methods based on the state distance, the state error and the average value of an imaginary mechanical quantity are analyzed. Finally, some numerical simulation experiments are studied.
Quantum Lyapunov control was developed in order to transform a quantum system from arbitrary initial states to a target state. The idea is to find control fields that steer the Lyapunov function to zero as $t\ensuremath{\rightarrow}\ensuremath{\infty}$, meanwhile the quantum system is driven to the target state. In order to shorten the time required to reach the target state, we propose two designs to optimize Lyapunov control in this paper. The first design makes the Lyapunov function decrease as fast as possible with a constraint on the total power of control fields, and the second design has the same purpose but with a constraint on each control field. Examples of a three-level system demonstrate that the evolution time for Lyapunov control can be significantly shortened, especially when high control fidelity is required. Besides, this optimal Lyapunov-based quantum control is robust against uncertainties in the free Hamiltonian and decoherence in the system compared to conventional Lyapunov control. We apply our optimal design to cool a nanomechanical resonator where a shorter cooling time is found with respect to the cooling time by the conventional Lyapunov design.
In this paper, we consider the fixed-time stabilization control problem of quantum systems modeled by Schrödinger equations. Firstly, the Lyapunov-based fixed-time stability criterion is extended to finite-dimensional closed quantum systems in the form of coherence vectors. Then for a two-level quantum system with single control input, a non-smooth fractional-order control law is designed using the relative state distance. By integrating the fixed-time Lyapunov control technique and the bi-limit homogeneity theory, the quantum system is proved to be stabilized to an eigenstate of the inherent Hamiltonian in a fixed time. Comparing with existing methods in quantum system control, the proposed approach can guarantee stabilization in a fixed time without depending on the initial states.
This paper presents a detailed Lyapunov-based theory to control and stabilize continuously-measured quantum systems, which are driven by Stochastic Schrodinger Equation (SSE). Initially, equivalent classes of states of a quantum system are defined and their properties are presented. With the help of equivalence classes of states, we are able to consider global phase invariance of quantum states in our mathematical analysis. As the second mathematical modelling tool, the conventional Ito formula is further extended to non-differentiable complex functions. Based on this extended Ito formula, a detailed stochastic stability theory is developed to stabilize the SSE. Main results of this proposed theory are sufficient conditions for stochastic stability and asymptotic stochastic stability of the SSE. Based on the main results, a solid mathematical framework is provided for controlling and analyzing quantum system under continuous measurement, which is the first step towards implementing weak continuous feedback control for quantum computing purposes.