多量子位Grover量子搜索算法的NMR仿真实现

核磁共振（NMR）技术目前是能有效实现量子计算的物理体系之一。多量子算符代数理论可以将幺正变换分解为一系列有限的单量子门和对角双量子门的组合。本文以核磁共振和多量子算符代数理论为基础,提出了实现多量子位Grover量子搜索算法的核磁共振脉冲序列设计方法,并在量子计算仿真程序上进行了3量子位的Grover量子搜索算法的实验验证。     

量子搜索算法体系的仿真实现及研究

核磁共振（NMR）技术被认为是最为有效的实现量子计算的物理体系之一。多量子算符代数理论可以将幺正变换分解为一系列有限的单量子门和对角双量子门的组合。本文以核磁共振和多量子算符代数理论为基础，提出了实现任意相位旋转角度的一般化量子搜索算法的核磁共振脉冲序列设计方法，并在量子计算仿真程序上进行了双量子位的不同相位旋转角度的量子搜索算法的实验验证。     

双线性系统反馈线性化鲁棒控制的新方法

研究了单输入的双线性系统的精确线性化及其鲁棒控制器设计问题,避免了用微分 几何的李导数、李括号的复杂运算,只用纯代数的方法给出了一般单输入双线性系统精确线 性化的充分条件.同时,在非线性不确定的条件下,讨论了双线性系统精确线性化的鲁棒控制 器的设计问题.     

基于光量子芯片的量子自旋链中完美态转移可编程量子模拟

近些年,量子计算物理实现技术进步很快,构建能够发挥实际用途的量子计算装置成为发展重点。采用量子模拟研究量子自旋系统的演化行为,相比于经典模拟会更加高效。一维量子自旋链中完美态转移模型在量子通信和量子计算领域具有重要的研究价值。提出一种基于双光子连续时间量子漫步的可编程完美态转移量子模拟方法,并且基于光量子芯片完成了2类特殊哈密顿量作用下XY型量子自旋链中双激发“周期-镜像”完美态转移的量子模拟实验,为模拟量子自旋系统的演化提供了一种实用且可扩展的实验方案。     

多量子位量子小波变换逻辑线路及仿真实现

由于量子计算相比于经典计算的突出优越性,量子小波变换的实现对于小波变换的理论完善和实际应用具有重要的意义,而逻辑线路是该变换实现的基础。应用多量子算符代数理论设计了3量子位Haar和D^（4）小波变换的逻辑线路,进而将逻辑线路转化成核磁共振系统可以实现的脉冲序列,并在量子计算仿真器（QCE）上进行了模拟实现,验证了逻辑线路的合理性。     

概率布尔控制网络的集可控

本文研究概率布尔控制网络的集可控性问题.首先,利用矩阵半张量积方法,得到概率布尔控制网络的代数表示.其次,借助一个新的算子构造不同的可控矩阵,进而通过可控矩阵考虑自由控制序列和网络输入控制下概率布尔控制网络的集可控性问题,得到了概率布尔控制网络集可控性的充要条件.最后,给出数值例子说明本文结果的有效性.     

基于恒定磁场的电子自旋量子比特系统任意量子态的最优制备

借助单量子位的Bloch球面表示,结合量子门实现量子态幺正演化的量子态调控机制,以恒定磁场为控制场,引入开关控制思想,提出了一种针对电子自旋量子系统任意量了态的最优制备策略.建立了量子系统及控制场的模型,并借助李群李代数,由经典最优控制的思想,获得任意量子态的最优制备.理论分析与仿真实验说明了该策略的优越性.     

网络化单连杆刚性关节机械臂系统的可控性

主要研究了网络化单连杆刚性关节机械臂系统的可控性问题.基于单输入—状态反馈线性化技术,提出了一种可行的网络化单连杆刚性关节机械臂系统可控性分析方法,并给出了网络化单连杆刚性关节机械臂系统可控性的一般准则.准则表明链式拓扑结构下的网络化单连杆刚性关节机械臂系统总是可控的,并且其控制输入能够精确地显式表达出来.最后通过仿真验证了所给理论结果的有效性.     

石墨烯量子点和量子比特

本文主要回顾了石墨烯量子点的制备以及基于石墨烯量子点自旋和电荷量子比特操作的研究进展,由于石墨烯材料相对较轻的原子重量使其具有较小的自旋轨道相互作用,另外含有核自旋的碳同位素13C在自然界中的含量大约只占1%,这使得超精细相互作用(即核自旋和电子自旋相互作用)较弱,所以石墨烯比其他材料具有较长的自旋退相干时间,在量子计算和量子信息中有非常好的应用前景.本文计算了5种静电约束制备的石墨烯量子点:1)扶手型单层石墨烯纳米条带,2)单层石墨烯圆盘,3)双层石墨烯圆盘,4)ABC堆积型三层石墨烯圆盘,5)ABA堆积型三层石墨烯圆盘.石墨烯量子点中自旋比特应用的关键是破坏谷简并,在1)中,主要是利用边界条件破坏谷简并,而2)–5)中是利用外磁场破坏谷简并.文章进一步介绍了自旋轨道相互作用和超精细相互作用对石墨烯量子点中自旋操作的影响.     

相互作用的量子系统模型及其物理控制过程

在充分考虑量子系统中粒子之间的相互作用以及可能需要的几何控制的基础上,建立了一个变量在李群的SU(4)上变化的、两个具有相互作用的自旋1/2粒子系统的数学模型.详细地描述了对具有相互作用的量子系统的物理控制过程.为进一步对量子系统有关的可控性、操纵以及反馈控制做好了准备工作.     

Controllability of spacecraft systems in a central gravitationalfield

The configuration space for rigid spacecraft systems in a central gravitational field can be modeled by SO(3)× IR³, where the special orthogonal group SO(3) represents the attitude dynamics and IR³ is for the orbital motion. The attitude dynamics of the spacecraft system is affected by the orbital elements through the well-known gravity-gradient torque. On the other hand, the effects of attitude-orbit coupling can also possibly be used to alter orbital motions by controlling the attitude. This controllability property is the primary issue of this paper. The control systems for spacecraft with either reaction wheels or gas jets being used as attitude controllers are proven in this study to be controllable. Rigorously establishing these results necessitates starting with the formal definitions of controllability and Poisson stability. It is then shown that if the drift vector field of the system is weakly positively Poisson stable and the Lie algebra rank condition is satisfied, controllability can be concluded. The Hamiltonian structure of the spacecraft model provides a natural route of verifying the property of weakly positive Poisson stability. Accordingly, the controllability is obtained after confirming the Lie algebra rank condition. Developing a methodology in deriving Lie brackets in the tangent space of T(SO(3)×IR³), i.e., the second tangent bundle is thus deemed necessary. A general formula is offered for the computation of Lie brackets of second tangent vector fields in TT(SO(3)^m×IRⁿ), in light of its importance in the fields of mechanics, robotics, optimal control, and nonlinear control, among others. With these tools, the controllability results can be proved. The analysis in this paper gives some insight into the attitude-orbit coupling effects and may potentially lead towards new techniques in designing controllers for large spacecraft systems     

Local controllability of nonlinear systems

The local controllability of control systems with an arbitrary number of controls is considered, first on an open set and then at a given point; necessary conditions are derived concerning the Lie algebra T′ generated by the input vector fields, and applied to gradient systems.The main results are a geometric sufficient condition for local controllability at a point, and an equivalent condition based on the computation of Lie brackets, assuming T′ to be (n − 1)-dimensional.     

Motion control of drift-free, left-invariant systems on Lie groups

In this paper we address the constructive controllability problem for drift-free, left-invariant systems on finite-dimensional Lie groups with fewer controls than state dimension. We consider small (ϵ) amplitude, low-frequency, periodically time-varying controls and derive average solutions for system behavior. We show how the pth-order average formula can be used to construct open-loop controls for point-to-point maneuvering of systems which require up to (p-1) iterations of Lie brackets to satisfy the Lie algebra controllability rank condition. In the cases p=2,3, we give algorithms for constructing these controls as a function of structure constants that define the control authority, i.e., the actuator capability, of the system. The algorithms are based on a geometric interpretation of the average formulas and produce sinusoidal controls that solve the constructive controllability problem with O(ϵ^P) accuracy in general (exactly if the Lie algebra is nilpotent). The methodology is applicable to a variety of control problems and is illustrated for the motion control problem of an autonomous underwater vehicle with as few as three control inputs     

Accessibility of bilinear networks of systems: control by interconnections

We analyze controllability properties for a class of bilinear interconnected systems, consisting of networks of linear systems, where the coupling parameters act as control variables. We characterize the system Lie algebra of the resulting bilinear control systems. Necessary and sufficient conditions for accessibility are derived in terms of the underlying interconnection graph. Our results generalize earlier work by Brockett on controllability of bilinear output feedback systems, as well as recent work of Costello and Egerstedt on the control of information-exchange networks for distributed computing.     

Controllability of quantum walks on graphs

In this paper, we consider discrete time quantum walks on graphs with coin, focusing on the decentralized model, where the coin operation is allowed to change with the vertex of the graph. When the coin operations can be modified at every time step, these systems can be looked at as control systems and techniques of geometric control theory can be applied. In particular, the set of states that one can achieve can be described by studying controllability. Extending previous results, we give a characterization of the set of reachable states in terms of an appropriate Lie algebra. Controllability is verified when any unitary operation between two states can be implemented as a result of the evolution of the quantum walk. We prove general results and criteria relating controllability to the combinatorial and topological properties of the walk. In particular, controllability is verified if and only if the underlying graph is not a bipartite graph and therefore it depends only on the graph and not on the particular quantum walk defined on it. We also provide explicit algorithms for control and quantify the number of steps needed for an arbitrary state transfer. The results of the paper are of interest in quantum information theory where quantum walks are used and analyzed in the development of quantum algorithms.     

The Hamiltonian formulation of tetrad gravity: Three-dimensional case

The Hamiltonian formulation of tetrad gravity in any dimension higher than two, using its first-order form where tetrads and spin connections are treated as independent variables, is discussed, and the complete solution of the three-dimensional case is given. For the first time, applying the methods of constrained dynamics, the Hamiltonian and constraints are explicitly derived and the algebra of Poisson brackets among all constraints is calculated. The algebra of Poisson brackets, among first-class secondary constraints, locally coincides with Lie algebra of the ISO(2,1) Poincaré group. All the first-class constraints of this formulation, according to the Dirac conjecture and using the Castellani procedure, allow us to unambiguously derive the generator of gauge transformations and find the gauge transformations of the tetrads and spin connections which turn out to be the same as found by Witten without recourse to the Hamiltonian methods [Nucl. Phys. B 311, 46 (1988)]. The gauge symmetry of the tetrad gravity generated by the Lie algebra of constraints is compared with another invariance, diffeomorphism. Some conclusions about the Hamiltonian formulation in higher dimensions are briefly discussed; in particular, that diffeomorphism invariance is not derivable as a gauge symmetry from the Hamiltonian formulation of tetrad gravity in any dimension where tetrads and spin connections are used as independent variables.     

Hamiltonian realizations of nonlinear adjoint operators

This paper addresses the issue of state-space realizations for nonlinear adjoint operators. In particular, the relationships between nonlinear Hilbert adjoint operators, Hamiltonian extensions and port-controlled Hamiltonian systems are established. Then, characterizations of the adjoints of controllability, observability and Hankel operators are derived from this analysis. The state-space realizations of such adjoint operators provide new insights on singular value analysis and duality issues in nonlinear control systems theory. Finally, a duality between the controllability and observability energy functions is proved.     

Motion planning for control-affine systems satisfying low-order controllability conditions

This paper is devoted to the motion planning problem for control-affine systems by using trigonometric polynomials as control functions. The class of systems under consideration satisfies the controllability rank condition with the Lie brackets up to the second order. The approach proposed here allows to reduce a point-to-point control problem to solving a system of algebraic equations. The local solvability of that system is proved, and formulas for the parameters of control functions are presented. Our local and global control design schemes are illustrated by several examples.     

Lie algebraic canonical representations in nonlinear control systems

This paper is the applied counterpart to previous results [5] for linear-analytic control systems. It is mainly concerned with two canonical representations of the exponential type. They exhibit the Lie algebraic structure of the system in such a form that results on weak controllability are easily derived in an algebraic manner. The first representation is a single exponential of a canonical Lie series in Hall's basis of the Lie algebra of vector fields. The second one is a factorization in terms of simpler exponentials of Hall's basic vectors. Both of them exhibit, as canonical coefficients, an infinite set of characteristic parameters which are a minimal representation of the input paths, when no drift occurs in the system (or, equivalently, in the weak control case). The weak controllability theorem is easily derived from these results, in a purely algebraic way.     

Accessibility of Switched Linear Systems

This note considers the controllability of switched linear systems. The structure of accessibility Lie algebra is revealed. Some accessibility properties are proved. Certain necessary and sufficient conditions for (local or global, weak or normal) controllability of a large class of switched linear systems are obtained.