Quantum Persistent Homology for Time Series
This work addresses a bottleneck for researchers analyzing time series data with quantum topological tools, though it is incremental as it builds on prior quantum algorithms.
The paper tackles the limitation of existing quantum persistent homology algorithms, which only handle point cloud data, by developing a quantum Takens's delay embedding algorithm to convert time series into point clouds, enabling topological feature extraction from time series using quantum methods.
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.