  Extracting useful information from large data sets can be a daunting task.
Topological methods for analyzing data sets provide a powerful technique for
extracting such information. Persistent homology is a sophisticated tool for
identifying such topological features -- connected components, holes, or voids
-- and for determining how such features persist as the data is viewed at
different scales. This paper provides quantum algorithms for calculating Betti
numbers in persistent homology, and for finding eigenvectors and eigenvalues of
the combinatorial Laplacian. The algorithms provide an exponential speedup over
classical algorithms for topological data analysis.
