Geometric Analysis and Optimal Control of Quantum Systems in the KP Configuration; Generalizations to nonlinear Systems with Symmetries

KP 配置中量子系统的几何分析和优化控制；

基本信息

批准号：
1710558
负责人：
Domenico D'Alessandro
金额：
$ 29.2万
依托单位：
Iowa State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-08-01 至 2022-10-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1710558&HistoricalAwards=false
关键词：
Geometric Analysis Optimal Control Quantum

项目摘要

The precise and efficient control of the state of quantum mechanical systems, such as atoms, nuclei and electrons, is a requirement in most applications of these systems. Such a control is typically obtained through the interaction with external, appropriately shaped, electromagnetic fields. Moreover, one often wants not only to drive the state to a desired value but also to optimize the available resources. The minimization of time is especially important. In applications to computation, fast dynamics result in the speed-up of the implemented algorithms. Furthermore, in general, the evolution has to occur within the time frame during which the mathematical model can be considered valid, before the effect of un-modeled dynamics becomes relevant. In this context, this research will accomplish three inter-related objectives: 1) It will provide explicit optimal control design algorithms for a large class of quantum mechanical systems very common in important applications. 2) It will provide an in depth mathematical analysis of the role of symmetries in quantum control systems and how these symmetries can be used to simplify mathematical models of quantum systems, thus considerably extending the existing theory. 3) It will validate the mathematical results through experiments in quantum optics and nuclear magnetic resonance via the collaboration with experimental laboratories. The overall result will be a rich toolbox for the optimal manipulation of quantum mechanical systems to be used in applications in secure communication, powerful quantum computing, design of measurement devices, medical diagnostics and, in general, every device which uses quantum systems. The activities will involve an interdisciplinary research team composed of engineers, physicists and mathematicians with the objective of developing a common language and science. The resulting knowledge will be the basis of a new area of engineering and a curriculum in Quantum Engineering, a field that will become very important in the future as the applications of quantum mechanics in everyday life continue to expand. The main mathematical tools used and developed in this research come from the field of differential geometry and in particular Riemannian and sub-Riemannian geometry. The starting point are the so-called KP mathematical models which are models whose state varies on a Lie group and whose dynamical equations correspond to a Cartan decomposition of the associated Lie algebra. The corresponding optimal control problems are, on one hand, very common in applications, and, on the other hand, explicitly solvable. In the project they serve as a test-bed to investigate properties of quantum control systems in general. These involve the role of symmetries, the qualitative behavior of optimal trajectories (geodesics) and the geometry of the reachable sets. A key technical ingredient of the mathematical approach is the use of symmetry reduction as a tool to analyze the control problem on a lower dimensional quotient space. On this space a simpler control problem can be posed and often explicitly solved. This procedure will substantially enlarge the existing toolbox in quantum control, which is frequently restricted to small dimensional systems. The experimental implementations of the resulting control design will be for systems in quantum optics and nuclear magnetic resonance, which are among the most promising candidates for the construction of quantum computers. Furthermore, the mathematical analysis will require the introduction of elements from the theory of singular and stratified spaces, which is important in other areas of applications of control besides quantum mechanics. In this context, this project will contribute to the development of control theory for classical systems as well.

在这些系统的大多数应用中，对量子机械系统状态（例如原子，核和电子等状态）的精确控制是必不可少的。这种对照通常是通过与外部，适当形状的电磁场的相互作用而获得的。此外，人们常常不仅希望将状态推向所需的价值，而且还希望优化可用资源。时间的最小化尤其重要。在计算应用程序中，快速动态导致了实现算法的加速。此外，通常，在数学模型可以视为有效的时间范围内，进化必须发生，然后在非模型动力学的效果变得相关。在这种情况下，这项研究将实现三个相互关联的目标：1）它将为在重要应用中非常常见的大型量子机械系统提供明确的最佳控制设计算法。 2）它将对量子控制系统中对称性的作用以及这些对称性如何使用来简化量子系统的数学模型，从而大大扩展现有理论的数学模型。 3）它将通过与实验实验室的合作进行量子光学和核磁共振的实验来验证数学结果。总体结果将是一个丰富的工具箱，用于最佳操纵量子机械系统，用于安全通信，强大的量子计算，测量设备的设计，医疗诊断，以及通常使用量子系统的每个设备。这些活动将涉及一个由工程师，物理学家和数学家组成的跨学科研究团队，目的是开发通用语言和科学。由此产生的知识将是新的工程领域和量子工程课程的基础，量子工程的课程将在将来变得非常重要，因为量子力学在日常生活中的应用不断扩大。这项研究中使用和开发的主要数学工具来自差异几何学领域，尤其是Riemannian和riemannian几何形状。起点是所谓的KP数学模型，该模型是其状态在谎言组中变化的模型，其动力学方程对应于相关谎言代数的cartan分解。一方面，相应的最佳控制问题在应用中非常普遍，另一方面，可以明确解决。在项目中，它们是研究量子控制系统的性质的测试床。这些涉及对称性的作用，最佳轨迹的定性行为（地球学）的定性行为和可触及集的几何形状。数学方法的关键技术成分是使用减少对称性的使用作为在较低维商空间上分析控制问题的工具。在这个空间上，可以提出更简单的控制问题并经常明确解决。此过程将大大扩大量子控制中现有的工具箱，该工具箱通常仅限于小维系统。所得控制设计的实验实现将用于量子光学和核磁共振的系统，这是构建量子计算机的最有希望的候选者之一。此外，数学分析将需要从奇异和分层空间理论引入元素，这在除量子力学以外的控制的其他领域很重要。在这种情况下，该项目也将有助于开发经典系统的控制理论。