Quantum Framework for Topological Data Analysis

Topological Data Analysis (TDA) methods combine tools from statistics and machine learning with concepts from algebraic topology usually with the purpose of classifying data based on its shape. These techniques provide advantages such as dimensionality reduction and resilience to noise. In addition, they can often recover information from signals or other complex dynamical systems that traditional methods fail to capture. In recent years, TDA methods have seen applications in many different classification problems from biology, materials science, robotics, and computer vision, among others. However, extracting topological features from a data set can be computationally expensive as in general it involves an NP problem. With the rapid development of quantum computers and their rise in popularity to deal with similar mathematical problems, it is natural to wonder if these new devices can provide an advantage for TDA. Indeed there have been several recent papers introducing quantum algorithms for TDA and discussing their potential to speedup or complement the current classical counterparts. This manuscript is a compilation of my works on the subject of quantum methods for TDA. My contributions include a novel quantum algorithm for persistent homology that is able to obtain topological features from a data set and track them through changes in resolution. In addition, the algorithm yields more information than other similar quantum algorithms and still has the potential to provide a quadratic speedup over classical counterparts. Furthermore, I introduce a subroutine that allows the aforementioned algorithm and similar ones to work with time series data sets like signals. Finally, I adapt techniques from quantum variational algorithms to estimate distances between persistence diagrams, so as to compare data sets through the topological features extracted by the persistent homology algorithm.