摘要
根据是否与环境等相互作用可将量子系统分为封闭和开放量子系统。本论文在封闭量子系统的研究中,完成了状态转移和轨迹跟踪控制律的设计任务以及收敛性证明。在开放量子系统中,实现了状态转移控制和量子门算符的制备与保持。
在封闭量子系统中,在状态转移研究方面,在液态核磁共振体系中引入辅助系统,采用李雅谱诺夫方法设计控制律,实现了核磁共振体系从一个高度混合态转移到一个有效纯态(伪纯态)的转移任务。在轨迹跟踪控制方面,利用相互作用绘景变换将对量子系统自然演化轨迹的跟踪问题转变成状态转移问题,或者通过引入误差变量将状态跟踪变成误差调节问题,在选用同样李雅谱诺夫函数的情况下,两种方法得到一样的控制律。在状态转移收敛性研究方面,针对自由哈密顿量能级差各不相等,各个能级之间可以直接耦合的理想系统,当目标轨迹是系统自然演化轨迹的情况下,在相互作用绘景中采用基于虚拟力学量的李雅谱诺夫函数,借助Barbalat引理分析动力学系统的收敛性。通过设计虚拟力学量具有非退化本征谱,将不变集缩小到只包含与虚拟力学量对易的状态；进一步,调节虚拟力学量本征值的取值来保证收敛。为了简化虚拟力学量的设计,论文中给出了虚拟力学量的一种固定的取法,并且证明了在所取值方式下,不变集中除目标态之外其他状态都是临界稳定的充要条件。对于更实际的非理想系统,当目标态是本征态或者是以对角密度矩阵表示的伪纯态时,证明了系统收敛到目标态的充要条件。
在开放量子系统的状态控制方面,论文分别对马尔科夫以及非马尔科夫开放量子系统进行了研究。在一个四能级马尔科夫开放量子系统中,借助无消相干技术,实现了对无消相干目标态的自由演化轨迹的跟踪控制,针对非理想系统,理论分析和系统仿真实验都显示了所设计的李雅谱诺夫控制律可以使系统收敛到目标态。在一个二能级非马尔科夫系统中,通过简化抵消漂移项的控制律的形式,避免了控制律奇异和状态转移路径的剧烈振荡的现象。在系统仿真实验过程中,设计一系列离散的目标纯态作为状态转移的中间路径,通过路径规划实现对状态轨迹的控制。
论文采用李雅普诺夫控制方法实现了非马尔科夫开放量子系统中单比特量子门算符的快速制备和保持。分析了控制律奇异的问题并提出有效解决方案。基于控制任务进行控制律改进,理论分析说明在改进的控制律作用下,系统一定能够到达目标算符,实现了高精度单量子比特门的制备,并且通过系统仿真实验证明了控制律对系统参数,如耦合强度等有很好的鲁棒性。相比于基于梯度下降脉冲设计(GRAPE)的最优控制方法,所设计的李雅谱诺夫控制具有速度快,目标门算符可以保持的优点,且在李雅谱诺夫函数选取合适的情况下,也能达到最优精度。
Quantum systems are grouped into closed systems and open systems according to whether the systems interact with their environment. The dissertation researches state control and operator manipulation in quantum systems. In closed systems, Lyapunov control laws for state transferring and state tracking are designed. Besides, a detailed convergence proof is accomplished. In open systems, the work focuses on complete state transferring and quantum gates operator implementation.
In closed quantum systems, state transferring and state tracking are considered. Firstly, state transferring is theoretically realized in liquid nuclear magnetic resonance (NMR) system. In this example, another quantum system, the so-called auxiliary system, is coupled to NMR. Lyapunov control law is designed to transfer NMR from a high-mixed initial thermal equilibrium state to an effective pure state (pseudo-pure state). Secondly, state tracking methods are discussed in detail. Error as a new controlled variable is introduced to change state tracking into state transferring. If the system's nature trajectory is tracked, interaction picture is also employed to change tracking into transferring. In this case, both the two methods produce the same control law if we choose the same Lyapunov function. Simulation experiments demonstrate that the tracking performance is related to target state and system structure. In the convergence proof, an ideal system, where energy differences between arbitrary levels of free Hamiltonian are different and the energy level is connected to each other, is investigated in interaction picture. To track the nature trajectory of the ideal controlled system, an average value of virtual observable quantity P is selected as Lyapunov function. The positive limit set of the controlled system is analyzed via Barbalat lemma. In the design process, P with non-degenerate eigen-spectrum is designed to shrink the limit set R to a smaller one, in which all the states in R commute with P; the eigenvalues of P is carefully selected to ensure convergence. To simplify the qualified P, the dissertation gives a candidate P, which is proved that the other states except the target one in the limit set under this P are critically stable. It is verified that a non-ideal system still converges to an eigen-state or pseudo-pure state described by diagonal density matrix, the sufficient and necessary condition of which is obtained.
The open quantum systems interacted with Markovian and Non-Markovian environment are investigated. The dissipative dynamics in open quantum system prohibits the complete state tracking. In a four-level Markovian system, decoherence-free subspace is employed to track a nature trajectory of a decoherence-free target state. That this non-ideal system under the designed control law converges to target state is verified by theoretical description and simulation experiments.
The Lyapunov control law is simplified to avoid oscillating state trajectory in a two-level Non-Markovian quantum system. Although a complete tracking is impossible except the case of equilibrium state tracking, a discrete target trajectory composed of pure states is implemented by path planning in simulation experiments.
The single quantum gates in Non-Markovian system is manipulated by Lyapunov control. The singularity problem arising in control law is studied and solved. A redesigned Lyapunov control law brings an great improvement on control effectiveness:1) it ensures that the system reach target operator;2) the operator preparation process is finished in a short time because of the monotonically decreasing of Lyapunov function. Compare with optimal control, the fast preparation process is obvious and the adaptive control law has a peculiar advantage that it keeps target operator unchanged for a significant time. Moreover, the optimal accuracy is also obtained by a proper Lyapunov function. In the simulation experiments, another benefit of the proposed control law is the nice robustness against the variations of system parameters.
引文
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