Quantum persistent homology
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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 sub-sampling 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 spectrum relates to that of the persistent combinatorial Laplacian, and thus allows us to recover the persistent Betti numbers which capture the persistent features of data. In addition, our algorithm can also extract the non-harmonic spectra of the Laplacian, which can be used for data analysis as well. We also test our algorithm on a point cloud data.