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Chinese Control Conference (CCC)

Feedback Control of Quantum System Based on State Distance

Kuang Sen, Shuang Cong

Published Aug 1, 2006• 0 citations
Publisher

Abstract

Based on Bures distance, a Lyapounov function that represents a desired state and an actual state is selected for quantum systems. Considering the problems that the initial state is and isn't orthogonal to the desired state respectively, a class of feedback control strategies is designed that ensures the stability of closed-loop systems. Finally, we make a simulation experiment on a spin-1/2 system with two types of controls and make an effect analysis of system evolution time and control size in the cases with different parameters.

DOI
10.1109/chicc.2006.280929

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Based on Bures distance, a Lyapounov function that represents a desired state and an actual state is selected for quantum systems. Considering the problems that the initial state is and isn't orthogonal to the desired state respectively, a class of feedback control strategies is designed that ensures the stability of closed-loop systems. Finally, we make a simulation experiment on a spin-1/2 system with two types of controls and make an effect analysis of system evolution time and control size in the cases with different parameters.
Abstract Based on Bures distance, a Lyapunov function that represents the distance between a desired state and the actual state of a quantum system is selected. Considering the cases that an initial state is and is not orthogonal to the desired state respectively, we propose a class of control strategies with state feedback that ensures the stability of the closed-loop control system. Especially, the asymptotic stability of the control system is analyzed, deduced and proved in detail. Finally, a simulation experiment on a spin-1/2 particle system is done and the relation between the system state evolution time and control value is analyzed with different parameters. Research results have general theoretical meaning for control of quantum systems.
In the design of complex quantum systems like ion traps for quantum computing, it is usually desired to stabilize a particular system state or make the system state track a desired trajectory. Several control theoretical approaches based on feedback seem attractive to solve such problems. But the uncertain dynamics introduced by measurement on quantum systems makes the synthesis of feedback control laws very complicated. Although we have not explicitly modeled the change in system dynamics due to measurement (we have assumed weak measurements), this is a first step towards a more detailed analysis and closed-loop feedback design. Here, we present a Lyapunov-based control approach on the lines of that developed by Mirrahimi, Rouchon, Turnici (2005). The states are assumed to be obtained from weak measurements. The Lyapunov control technique has not been applied to realistic quantum systems so far. We have extended and applied the technique to two realistic physical systems - the quantum harmonic oscillator and the n-qubit system. We also propose to extend this concept to ion traps.
The paper proposes a gradient method for feedback control and stabilization of quantum systems using Schröodinger’s and Lindblad’s descriptions. The eigenstates of the quantum system are defined by the spin model. First, a gradient-based control law is computed using Schröodinger’s description. Next, an estimate of state of the quantum system is obtained using Lindblad’s differential equation. In the latter case, by applying Lyapunov’s stability theory and LaSalle’s invariance principle one can compute a gradient control law which assures that the quantum system’s state will track the desirable state within acceptable accuracy levels. The performance of the control loop is studied through simulation experiments for the case of a two-qubit quantum system.
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, a Lyapunov-based feedback control (LFC) is proposed for the state transfer of a two-level stochastic quantum system. The quantum state is described in the Cartesian coordinate system. An exponential function is selected as the Lyapunov function to ensure the stability of the system. The Lyapunov-based feedback control strategy is designed according to the stochastic Lyapunov stability theorem. Numerical simulations results show that the proposed LFC strategy has the ability to transfer the stochastic system from any initial mixed state to the desired eigenstate, and the LFC has high convergence rate and high state transfer fidelity.