Quantum Algorithm for Topological Data Analysis

Physiological signals are typically nonstationary, noisy, and nonlinear, and current signal processing methods may fail due to underlying assumptions. Persistent homology is a powerful mathematical tool that can be used to extract useful information from large datasets, including topological features and how these features persist or change at different scales. However, the computation is often a rather formidable task. I present a quantum algorithm which yields a summary of topological features by computing the eigenvectors and eigenvalues of the persistent combinatorial Laplacian and the persistent Betti numbers of a point cloud. In particular, the algorithm tracks how topological features of a point cloud, such as the number of connected components, holes, and voids, change across different resolutions or scales.