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Control theory & applications

State manipulation in Lyapunov-based quantum system control methods

Shuang Cong

Published Jan 1, 2012• 3 citations
OpenAlex

Abstract

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.

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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.
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.
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.
A Lyapunov-based method is presented for stabilizing and controlling of closed quantum systems. The proposed method is constructed upon a novel quantum Lyapunov function of the system state trajectory tracking error. A positive-definite operator in the Lyapunov function provides additional degrees of freedom for the designer. The stabilization process is analyzed regarding two distinct cases for this operator in terms of its vanishing or non-vanishing commutation with the Hamiltonian operator of the undriven quantum system. To cope with the global phase invariance of quantum states as a result of the quantum projective measurement postulate, equivalence classes of quantum states are defined and used in the proposed Lyapunov-based analysis and design. Results show significant improvement in both the set of stabilizable quantum systems and their invariant sets of state trajectories generated by designed control signals. The proposed method can potentially be applied for high-fidelity quantum control purposes in quantum computing frameworks.
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.
The condition of a quantum Lyapunov-based control which can be well used in a closed quantum system is that the method can make the system convergent but not just stable. In the convergence study of the quantum Lyapunov control, two situations are classified: nondegenerate cases and degenerate cases. For these two situations, respectively, in this paper the target state is divided into four categories: the eigenstate, the mixed state which commutes with the internal Hamiltonian, the superposition state, and the mixed state which does not commute with the internal Hamiltonian. For these four categories, the quantum Lyapunov control methods for the closed quantum systems are summarized and analyzed. Particularly, the convergence of the control system to the different target states is reviewed, and how to make the convergence conditions be satisfied is summarized and analyzed.
One main problem of quantum control in quantum systems is the manipulation of states including state transfer and state tracking. We design control fields to realize arbitrary state transfer for open quantum system by Lyapunov stability theory, and investigate the state transfer for non-Markovian system with phase relaxation and energy dissipative relaxation. The numerical simulations illustrate that arbitrary state (eigenstate, superposition state or mixed state) transfer and maintenance for non-Markovian system can be realized under Lyapunov control function by an external steady control field of proper amplitude, with a success rate of more than 98%. When the right control field and function are implemented, not only the decoherence is compensated completely but also the purity of quantum states is maintained in the process of state transfer.