Pade approximation, noise filtering, and quantum state transfer

Padé近似在自然科学和数学中有许多应用，它们被用来近似特殊函数的值。该项目将开发机械和技术，使其可用于引力波探测、应用于核废料的核磁共振光谱、脑/乳腺癌探测和石油探测。 该项目的另一个方面是研究与量子信息和量子计算机问题有关的所有这些对象。Padé逼近的历史可以追溯到Charles Hermite证明欧拉数是超越数。Hermite的博士生Henri Padé系统地扩展了这些技术。Padé逼近是一个有理函数，其分子和分母的次数分别为n和m，并且关于特定点的幂级数展开与给定函数的幂级数展开一致，直到第（n+m）项。Padé逼近的许多吸引人的特点之一是它们的快速收敛性和全局收敛现象。Padé逼近的行为中存在一些陷阱和许多有趣的开放问题，这些问题将在本项目中进行研究。PI计划培训研究生，并通过出版物、arXiv预印本和会议公开传播研究结果。该项目的主要目标的第一部分包括为基于Padé逼近的去噪方案开发数学基础，该方案最近由物理学家丹尼尔贝西斯和卢卡佩罗蒂提出。噪声滤波方法有很多种，但大多数经典方法在信噪比接近1时失效。Bessis-Perotti方法在几种情况下都是有效的。基本的数学问题在于分析理性扰动下Padé逼近的极点的行为。PI证明了马尔可夫函数有理扰动的Padé逼近的一些收敛结果，这些结果将进一步扩展并适应于去噪方案。正交多项式和雅可比矩阵与Padé逼近密切相关，并且在它们自己的权利中被广泛使用。PI打算探索和使用一些最近的渐近公式的非经典正交多项式的单位圆上的布赖恩Simanek和PI的噪声滤波方法。主要目标的第二部分包括进一步研究1D链中量子态转移与图上自旋构型之间的关系，这是最近由Gerald Dunne，Gamal Mograby，Sasha Tehraev和PI提出的。PI还计划使用雅可比矩阵和正交多项式的理论，找到一种设计具有非最近邻相互作用的一维链的系统方法，并使其适用于某些图形的情况。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Padé approximations have many applications in natural sciences and mathematics where they are used to approximate values of special functions. This project will develop the machinery and techniques so they can be used for gravitational wave detection, nuclear magnetic resonance spectroscopy as applied to nuclear waste, brain/breast cancer detection, and oil detection. Another aspect of the project is to study all these objects in relation to problems of quantum information and quantum computers. The history of Padé approximants goes back to Charles Hermite’s proof that Euler's number is transcendental. Henri Padé, a doctoral student of Hermite, systematically extended these techniques. A Padé approximant is a rational function with the degrees of numerator and denominator n and m, respectively, and the power series expansion about a specific point agreeing with the power series expansion of the given function up to the (n+m)-th term. One of many attractive features of Padé approximants is their fast convergence and the phenomenon of global convergence. There are some pitfalls in the behavior of Padé approximants and many interesting open problems, which will be studied in this project. The PI plans to train graduate students and disseminate results publicly through publications, arXiv preprints, and at conferences.The first part of primary goals of this project includes developing the mathematical foundation for a denoising scheme based on Padé approximants, which was recently proposed by physicists Daniel Bessis and Luca Perotti. There are many noise filtering methods available, but most of the classical ones fail when the signal-to-noise ratio approaches 1. The Bessis-Perotti method has been shown to be computationally effective in several cases. The underlying mathematical problem consists in the analysis of behavior of poles of Padé approximants under rational perturbations. The PI proved some convergence results for Padé approximants of rational perturbations of Markov functions, which will be further extended and adapted to the denoising scheme. Orthogonal polynomials and Jacobi matrices are intimately related to Padé approximants, and are widely used tools in their own rights. The PI intends to explore and to use some recent asymptotic formulas for nonclassical orthogonal polynomials on the unit circle of Brian Simanek and the PI in relation to the noise filtering method. The second part of primary goals includes further investigations of the relation between quantum state transfers in 1D chains and in spin configurations on graphs, which was recently proposed by Gerald Dunne, Gamal Mograby, Sasha Teplyaev, and the PI. The PI also plans to use the theory of Jacobi matrices and orthogonal polynomials to find a systematic approach to designing 1D chains with non-nearest neighbor interactions and to adapt it to the case of some graphs.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.