Quantum algorithms for topological and geometric analysis of data

Extracting useful information from large data sets can be a daunting task. Topological methods for analysing data sets provide a powerful technique for extracting such information. Persistent homology is a sophisticated tool for identifying topological features and for determining how such features persist as the data is viewed at different scales. Here we present quantum machine learning algorithms for calculating Betti numbers—the numbers of connected components, holes and voids—in persistent homology, and for finding eigenvectors and eigenvalues of the combinatorial Laplacian. The algorithms provide an exponential speed-up over the best currently known classical algorithms for topological data analysis. Persistent homology allows identification of topological features in data sets, allowing the efficient extraction of useful information. Here, the authors propose a quantum machine learning algorithm that provides an exponential speed up over known algorithms for topological data analysis.