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International Conference on Natural Computation (ICNC)

Quantum System Control Based on Lyapunov Technology

Qian Zhang, Wei Wang, Lan Wang

Published Jan 1, 2008• 0 citations
Publisher

Abstract

This paper designed a quadratic Lyapunov function, and obtained the control law for the controlled system via Lyapunov stability theorem. The convergence of the controlled system was analyzed. The simulation experiment on a spin-1/2 particle system was done, which shown the validity of the method studied in this paper. The relation between the value of the parameter in the control law and the corresponding control effect was also obtained.

DOI
10.1109/icnc.2008.412

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This paper designed a quadratic Lyapunov function, and obtained the control law for the controlled system via Lyapunov stability theorem. The convergence of the controlled system was analyzed. The simulation experiment on a spin-1/2 particle system was done, which shown the validity of the method studied in this paper. The relation between the value of the parameter in the control law and the corresponding control effect was also obtained.
The unitary matrix was constructed by using eigenstates of the controlled quantum system so as to perform a coordinate rotation on the controlled system.In such a way the local phases of all states of the system were eliminated.Then the error between the controlled state and the target state after transformation was selected as the Lyapunov function,and the system control law was designed based on Lyapunov stability theorem with the pre-condition that the system is kept stable.The detailed design procedure of the controller needed for preparation of the system states and the comparison analysis of the results were given.Finally,the system simulation experiments on a spin-1/2 particle system were implemented to prepare both the eigen and superposition target states with the proposed control method,and the relation between the system state evolution time and control value was analyzed with different parameters.
By means of the design method for bilinear systems in the macroscopic field in association with the characteristic that the mathematical model of pure state quantum system is bilinear,and of selecting the performance index of the controlled system,the optimal control strategy in quantum systems was proposed based on the Lyapunov stability theorem.The detailed design process of optimal control law was derived.Some simulation experiments were performed on a spin-1/2 particle system.Experimental results demonstrate the effectiveness of the method proposed.At the same time,the influence of different parameters on the system control effects is analyzed.
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.
By using a quadratic Lyapunov function with un-known parameters and a particle swarm optimization (PSO) algorithm, this paper proposes a new Lyapunov control scheme for quantum systems. This approach can achieve high-probability population transfer to a given target state. For the case when the target state is an eigenstate of the internal Hamiltonian, we introduce a virtual control law into the system model and design corresponding control law. The stability of the system under the action of the Lyapunov control law is analyzed via the LaSalle invariance principle. For the case when the target state is a superposition state, we design a control law by performing a unitary transformation for the quantum system model under consideration. To achieve desired state transfer, we further introduce a PSO algorithm to search for the unknown parameters contained in the control law. Numerical results on a five-level quantum system and a three-qubit system are presented to demonstrate the effectiveness of the proposed approaches.
Theoretical study on the control of spin-1/2 quantum system is an important matter for the development of quantum control theory. Lyapunov methods play vital role in quantum control because of their concise mathematical rigor. Previously, using the Lyapunov methods, it has been shown that signum function control law can be employed to control spin-1/2 quantum system to achieve NOT quantum gate operation. This paper demonstrates that other forms of control law can be used to control spin-1/2 quantum system to achieve the same NOT quantum gate operation. Two examples of those other forms are provided, namely hyperbolic tangent (tanh) function and proportional multiplier. It is also explained why signum function is not the only function that can be used to control spin-1/2 quantum system. This paper also shows the advantage of using fully-differentiable control law instead of signum function, which is not fully-differentiable. Control law design and stability analysis are conducted using Lyapunov methods. The numerical simulation results show that tanh and proportional multiplier function can also control spin-1/2 quantum system and achieve the NOT quantum gate operation.
We investigate in depth the Lyapunov-based quantum control methods including the relation between the Lyapunov-based control and the optimal control,the Lyapunov function and the performance index,the range of application and the existing problems.Combining our results with intrinsic properties of quantum system,we develop several control schemes with improvements for different objectives such as the regulation of eigenstates,superposition state or mixed state.The design ideas,the physical significance and the range of suitable applications are analyzed.The designs method employing Lyapunov stability theorem in controlling quantum system is systemized.