Quantum Distance Approximation for Persistence Diagrams

Topological Data Analysis methods can be useful for classification and clustering tasks in many different fields as they can provide two dimensional persistence diagrams that summarize important information about the shape of potentially complex and high dimensional data sets. The space of persistence diagrams can be endowed with various metrics such as the Wasserstein distance which admit a statistical structure and allow to use these summaries for machine learning algorithms. However, computing the distance between two persistence diagrams involves finding an optimal way to match the points of the two diagrams and may not always be an easy task for classical computers. In this work we explore the potential of quantum computers to estimate the distance between persistence diagrams, in particular we propose variational quantum algorithms for the Wasserstein distance as well as the $d^{c}_{p}$ distance. Our implementation is a weighted version of the Quantum Approximate Optimization Algorithm that relies on control clauses to encode the constraints of the optimization problem.