Numerical Methods in Noncommutative Matrix Analysis

该项目将开发对现代物理学和量子信息有用的算法。特别是，所开发的方法对于建模适用于光子芯片和量子电路的拓扑激光器将是重要的。这些应用对计算科学提出了挑战，特别是对线性代数算法，因此它们可以在多变量设置下与联合测量一起工作，并在识别存在的理论极限的情况下与不可公度可观测对象一起工作。该项目将考虑各种多变量线性代数算法，主要是那些满足计算量子物理和量子信息的迫切需要的算法，但也包括那些可以提高计算机形状分析的速度和精度的算法。该项目将有学生参与，并提供跨学科项目的培训。这个项目将为量子物理和图像分析中出现的矩阵和算符的集合开发数值方法。这包括与无限维系统的有限维近似一起工作的算法，导致对准晶、非晶态系统和具有缺陷的周期系统进行更好的计算机建模。将开发的方法和算法将在包括周期驱动系统在内的拓扑绝缘子的研究中有用。该项目将研究各种形式的光谱，包括局域态密度的变化，这是许多物理和化学领域的标准工具。关于K理论的预期工作有望产生新的更好的工具，可供理论物理学家在数值研究中使用。这些方法的核心是联合近似特征向量的研究。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The project will develop algorithms that are useful for modern physics and quantum information. In particular, the methods developed will be important for modeling topological lasers which are applicable to photonic chips and quantum circuitry. These applications pose challenges to the computational science and in particular to the linear algebra algorithms so they can work with joint measurement in multivariable setting and incommensurate observables recognizing the theoretical limits that exist. The project will consider a variety of multivariable linear algebra algorithms, mainly those filling an immediate need in computational quantum physics and quantum information, but also those that can increase the speed and accuracy of computer shape analysis. The project will involve students and provide training in interdisciplinary projects. This project will develop numerical methods for collections of matrices and operators arising in quantum physics and image analysis. This includes algorithms that work with finite-dimensional approximations to infinite-dimensional systems, leading to better computer modeling of quasicrystals, amorphous systems and periodic systems with defects. Methods and algorithms to be developed will be useful in the study of topological insulators, including periodically-driven systems. The project will study various forms of spectra, including variations of the local density of states, a standard tool in many areas of physics and chemistry. The anticipated work on K-theory is expected to produce new and better tools that can be used by theoretical physicists in numerical studies. At the core of these methods is the study of joint approximate eigenvectors.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.