Quantum Persistent Homology for Time Series

TL;DR

This paper introduces a quantum approach to persistent homology for time series data, transforming time series into point clouds via quantum Takens's embedding, enabling efficient topological analysis.

Contribution

It develops a quantum Takens's delay embedding algorithm to convert time series into point clouds for quantum persistent homology analysis.

Findings

Quantum Takens's embedding effectively converts time series into point clouds.

Quantum persistent homology can extract topological features from time series.

The approach potentially reduces computational complexity for topological data analysis.

Abstract

Persistent homology, a powerful mathematical tool for data analysis, summarizes the shape of data through tracking topological features across changes in different scales. Classical algorithms for persistent homology are often constrained by running times and memory requirements that grow exponentially on the number of data points. To surpass this problem, two quantum algorithms of persistent homology have been developed based on two different approaches. However, both of these quantum algorithms consider a data set in the form of a point cloud, which can be restrictive considering that many data sets come in the form of time series. In this paper, we alleviate this issue by establishing a quantum Takens's delay embedding algorithm, which turns a time series into a point cloud by considering a pertinent embedding into a higher dimensional space. Having this quantum transformation of time series to point clouds, then one may use a quantum persistent homology algorithm to extract the topological features from the point cloud associated with the original times series.