  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.
