Quantum Persistent Homology

Abstract: 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.