论文标题
多参数持续的同源性结构和量子计算
Multiparameter Persistent Homology-Generic Structures and Quantum Computing
论文作者
论文摘要
以下文章是在拓扑数据分析中的多参数持续同源性研究中的交换代数的应用。特别是，多项式环上模块有限分辨率的理论应用于多参数持续模块。使用跨越多数数十年研究的结果研究的结果研究了此类分辨率和所涉及的分类空间的通用结构，首先是研究由Buchsbaum和Eisenbud推广的自由决议的通用结构特性。使用计算机代数软件包MACAULAY2以及用于计算的代码进行了许多明确的计算。本文是交换代数的理论结果的集合，这对于将来使用Gröbner基础，标准单项理论，Young Tableaux，Schur Founctors和Schur多项式以及经典代表理论和不变理论涉及的线性代数组中的理论是必不可少的基础。所使用的方法是一般特征免费的，旨在在整数上使用，以便对数据科学的应用和计算有用。作为一个应用，我们解释了如何应用2参数持续的同源性来研究与二次汉密尔顿人相关的时变互动图，例如Ising模型或Kitaev的Torus代码和其他表面代码。
The following article is an application of commutative algebra to the study of multiparameter persistent homology in topological data analysis. In particular, the theory of finite free resolutions of modules over polynomial rings is applied to multiparameter persistent modules. The generic structure of such resolutions and the classifying spaces involved are studied using results spanning several decades of research in commutative algebra, beginning with the study of generic structural properties of free resolutions popularized by Buchsbaum and Eisenbud. Many explicit computations are presented using the computer algebra package Macaulay2, along with the code used for computations. This paper serves as a collection of theoretical results from commutative algebra which will be necessary as a foundation in the future use of computational methods using Gröbner bases, standard monomial theories, Young tableaux, Schur functors and Schur polynomials, and the classical representation theory and invariant theory involved in linear algebraic group actions. The methods used are in general characteristic free and are designed to work over the ring of integers in order to be useful for applications and computations in data science. As an applications we explain how one could apply 2-parameter persistent homology to study time-varying interactions graphs associated to quadratic Hamiltonians such as those in the Ising model or Kitaev's torus code and other surface codes.
