title: Quantum algorithms for topological and geometric analysis of data.
abstract: 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.
authors.affiliation: Department of Mechanical Engineering, Research Lab for Electronics, Massachusetts Institute of Technology, MIT 3-160, Cambridge, Massachusetts 02139, USA.
authors.forename: Seth
authors.initials: S
authors.lastname: Lloyd
authors.affiliation: Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1.
authors.forename: Silvano
authors.identifier: http://orcid.org/0000-0002-0252-6108
authors.initials: S
authors.lastname: Garnerone
authors.affiliation: Department of Physics and Astronomy, Center for Quantum Information Science &Technology, University of Southern California, Los Angeles, California 90089-0484, USA.
authors.forename: Paolo
authors.initials: P
authors.lastname: Zanardi