Multiparameter Persistent Homology-Generic Structures and Quantum Computing

TL;DR

This paper applies commutative algebra to analyze multiparameter persistent homology, exploring generic structures of resolutions, with computational tools, and discusses potential applications in quantum computing and data science.

Contribution

It introduces a theoretical framework connecting commutative algebra with multiparameter persistent homology, including explicit computations and applications to quantum systems.

Findings

Explicit computations using Macaulay2

Characterization of generic structures of resolutions

Potential applications in quantum computing and data analysis

Abstract

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 Gr\"obner 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.