  Persistent homology is a powerful mathematical tool that summarizes useful
information about the shape of data allowing one to detect persistent
topological features while one adjusts the resolution. However, the computation
of such topological features is often a rather formidable task necessitating
the subsampling the underlying data. To remedy this, we develop an efficient
quantum computation of persistent Betti numbers, which track topological
features of data across different scales. Our approach employs a persistent
Dirac operator whose square yields the persistent combinatorial Laplacian, and
in turn the underlying persistent Betti numbers which capture the persistent
features of data. We also test our algorithm on point cloud data.
