Quantum algorithms for topological and geometric analysis of big data

Abstract

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.

20 pages, plain TeX

References in corpus (3)

• Quantum principal component analysis
• Quantum algorithms for supervised and unsupervised machine learning
• Algorithms in Real Algebraic Geometry: A Survey

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• A Quantum Implementation Model for Artificial Neural Networks
• Bifurcations, time-series analysis of observables, and network properties in a tripartite quantum system
• The Learnability of Unknown Quantum Measurements
• Quantum Machine Learning For Classical Data