Quantum Persistent Homology for Time Series

Persistent homology, a powerful mathematical tool for data analysis, summarizes the shape of data through tracking changes to topological features across 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, like biological signals and stock prices, come in the form of time series. We alleviate this issue by establishing a quantum Takens's delay embedding algorithm, which identifies a time series with a point cloud by considering an embedding into its phase space. Having this quantum transformation of time series to point clouds, one may then use a quantum persistent homology algorithm to extract the topological features from the point cloud associated with the original times series. Furthermore, embeddings retain all topological information, so the topological features extracted from the point cloud can be used to analyse the corresponding time series.