马尔可夫开放量子系统无消相干子空间和目标态的稳定化控制

发布时间：2018-08-19 11:58

【摘要】：近年来,量子信息技术的发展十分迅速。在进行量子信息处理的过程中,环境引起的耗散或消相干效应会使所考虑的量子系统成为开放系统。对于开放量子系统而言,其动力学演化不再是幺正的,这将导致存储在系统中的量子信息发生错误甚至丢失。研究开放量子系统的控制方法,使其尽可能免受消相干的影响,同时实现系统状态的高效操纵,对于量子信息技术的发展具有重要意义。论文针对马尔可夫开放量子系统,研究基于李雅普诺夫方法的消相干控制和基于哈密顿量设计的状态控制,主要内容包括以下几个方面:1)概述量子控制的起源和发展,重点介绍开放量子系统控制的国内外研究现状,包括开放量子系统的消相干控制和状态控制。2)马尔可夫开放量子系统无消相干子空间的快速李雅普诺夫控制。无消相干子空间是量子系统进行幺正演化的所有量子态张成的一个子空间,将系统状态驱动至无消相干子空间中是一种有效避免消相干的方法。论文借助李雅普诺夫方法,提出了一个将马尔可夫开放量子系统快速驱动至无消相干子空间的控制方案。在该方案中,我们首先设计基于李雅普诺夫方法的控制律,特别是要设计具有快速性的近似棒棒控制律。然后,从数学上严格证明控制场作用下整个系统对于无消相干子空间的收敛性。在此基础上,通过分析系统的LaSalle最大不变集与无消相干子空间之间的关系,获得使系统对于无消相干子空间快速收敛的控制哈密顿量的构造方法。最后,在一个三维开放量子系统上进行仿真实验,以便验证方案的控制效果。3)开放量子系统的状态收敛控制。基于哈密顿量的构造,论文提出了一个使得开放量子系统收敛至目标态的哈密顿控制方法。首先,针对一个耗散项已知不变的开放量子系统,求解含有未知哈密顿量的系统方程以便得到系统的稳定解,通过将稳定解与期望目标态进行比较构造出哈密顿量的形式。其次,借助相干矢量体系将原系统的非线性模型转化为仿射线性模型,继而对后者的稳定性进行分析,由此导出原系统的稳定性条件。最后,我们在一个两能级系统上进行仿真实验,以验证所提出的哈密顿控制方案的有效性。
[Abstract]:In recent years, quantum information technology has developed rapidly. In the process of quantum information processing, the dissipative or decoherence effect caused by environment will make the quantum system under consideration as an open system. For open quantum systems, the dynamical evolution is no longer unitary, which will lead to errors or even loss of quantum information stored in the system. It is of great significance for the development of quantum information technology to study the control method of open quantum system so that it can avoid the influence of decoherence as much as possible and realize the efficient manipulation of system state at the same time. In this paper, for Markov open quantum systems, we study the decoherence control based on Lyapunov method and the state control based on Hamiltonian design. The main contents include the following aspects: 1) Overview of the origin and development of quantum control. The research status of open quantum system control at home and abroad is introduced, including the decoherence control and state control of open quantum system. 2) the fast Lyapunov control of Markov open quantum system without coherent subspace is introduced. Non-decoherence subspace is a subspace of Zhang Cheng of all quantum states in which the quantum system is unitary. Driving the state of the system to the non-decoherence subspace is an effective method to avoid decoherence. With the help of Lyapunov method, a control scheme is proposed in which Markov open quantum system is rapidly driven to a non-coherent subspace. In this scheme, we first design the control law based on Lyapunov method, especially the approximate rod control law with rapidity. Then, the convergence of the whole system to the non-coherent subspace under the action of the control field is proved mathematically. On this basis, by analyzing the relationship between the maximum LaSalle invariant set of the system and the non-coherent subspace, a construction method of the control Hamiltonian is obtained to make the system converge rapidly for the non-decoherence subspace. Finally, simulation experiments are carried out on a three-dimensional open quantum system to verify the control effect of the proposed scheme. 3) the state convergence control of the open quantum system. Based on the construction of Hamiltonian, a Hamiltonian control method is proposed to converge the open quantum system to the target state. Firstly, for an open quantum system with known invariant dissipative terms, the system equation with unknown Hamiltonian is solved to obtain the stable solution of the system, and the form of Hamiltonian is constructed by comparing the stable solution with the desired target state. Secondly, the nonlinear model of the original system is transformed into an affine linear model by means of the coherent vector system, and then the stability of the latter is analyzed, and the stability conditions of the original system are derived. Finally, simulation experiments are carried out on a two-level system to verify the effectiveness of the proposed Hamiltonian control scheme.
【学位授予单位】：中国科学技术大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O413

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