Pade approximation, noise filtering, and quantum state transfer
Pade 近似、噪声过滤和量子态转移
基本信息
- 批准号:2008844
- 负责人:
- 金额:$ 21.07万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Standard Grant
- 财政年份:2020
- 资助国家:美国
- 起止时间:2020-07-01 至 2024-06-30
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
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.
逼近法在自然科学和数学中有许多应用,它们被用来逼近特殊函数的值。该项目将开发用于引力波探测、核废料核磁共振波谱检测、脑癌/乳腺癌检测和石油检测的机器和技术。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相关的所有这些对象。帕德帕尔近似的历史可以追溯到查尔斯·赫米特证明欧拉数是超越的。Hermite的博士生Henri pad<s:1>系统地扩展了这些技术。帕德帕尔近似是分子和分母的度数分别为n和m的有理函数,其关于某一点的幂级数展开式与给定函数的幂级数展开式一直到第(n+m)项一致。pad<s:1>近似算法具有快速收敛和全局收敛的特点。在pad<s:1>近似算法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结果。该项目的第一部分主要目标包括为基于pad<s:1>近似值的去噪方案开发数学基础,该方案最近由物理学家Daniel Bessis和Luca Perotti提出。现有的噪声滤波方法很多,但当信噪比接近1时,经典的滤波方法大多失效。Bessis-Perotti方法在一些情况下被证明是计算有效的。潜在的数学问题在于分析在有理摄动下帕德帕尔近似的极点的行为。该方法证明了马尔可夫函数有理摄动的pad<s:1>近似的一些收敛结果,并将其进一步推广和适应于去噪方案。正交多项式和雅可比矩阵与帕德瓦近似密切相关,它们本身就是广泛使用的工具。本文旨在探讨和应用Brian Simanek单位圆上的非经典正交多项式的一些最新的渐近公式,以及PI与噪声滤波方法的关系。主要目标的第二部分包括进一步研究一维链中的量子态转移和图上自旋构型之间的关系,这是最近由Gerald Dunne, Gamal Mograby, Sasha Teplyaev和PI提出的。PI还计划使用雅可比矩阵和正交多项式理论来找到一种系统的方法来设计具有非最近邻相互作用的一维链,并使其适应某些图的情况。该奖项反映了美国国家科学基金会的法定使命,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。
项目成果
期刊论文数量(7)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Perfect quantum state transfer on diamond fractal graphs
- DOI:10.1007/s11128-020-02828-w
- 发表时间:2019-09
- 期刊:
- 影响因子:2.5
- 作者:Maxim S. Derevyagin;G. Dunne;Gamal Mograby;A. Teplyaev
- 通讯作者:Maxim S. Derevyagin;G. Dunne;Gamal Mograby;A. Teplyaev
Spectra of perfect state transfer Hamiltonians on fractal-like graphs
- DOI:10.1088/1751-8121/abc4b9
- 发表时间:2020-03
- 期刊:
- 影响因子:0
- 作者:Gamal Mograby;Maxim S. Derevyagin;G. Dunne;A. Teplyaev
- 通讯作者:Gamal Mograby;Maxim S. Derevyagin;G. Dunne;A. Teplyaev
A theorem of Joseph-Alfred Serret and its relation to perfect quantum state transfer
约瑟夫-阿尔弗雷德·塞雷特定理及其与完美量子态转移的关系
- DOI:10.1016/j.exmath.2020.12.001
- 发表时间:2021
- 期刊:
- 影响因子:0.7
- 作者:Derevyagin, Maxim;Minenkova, Anastasiia;Sun, Nathan
- 通讯作者:Sun, Nathan
Connection coefficients for ultraspherical polynomials with argument doubling and generalized bispectrality
具有参数加倍和广义双谱性的超球形多项式的连接系数
- DOI:10.1007/s11854-023-0271-6
- 发表时间:2023
- 期刊:
- 影响因子:0
- 作者:Derevyagin, Maxim;Geronimo, Jeffrey S.
- 通讯作者:Geronimo, Jeffrey S.
Complex Jacobi matrices generated by Darboux transformations
由达布变换生成的复雅可比矩阵
- DOI:10.1016/j.jat.2023.105876
- 发表时间:2023
- 期刊:
- 影响因子:0.9
- 作者:Bailey, Rachel;Derevyagin, Maxim
- 通讯作者:Derevyagin, Maxim
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