Quantum Distance Approximation for Persistence Diagrams

Topological data analysis (TDA) methods aim to characterize data shape using topological properties. Since these features are invariant under continuous transformations like rotations these methods have found numerous applications to different fields, from biology [1-6] to chemistry and materials science [7-11]. They have been used to solve problems like classification and clustering of images for action recognition [12], handwriting analysis [13], and even classification and clustering of signals [1, 12, 14-16]. With the recent rise of quantum computers, a few works have appeared exploring the potential advantages they can provide for TDA methods [17-21]. TDA methods first extract topological features from the data - such as the number of connected components, holes, and voids - via persistent homology, tracking them across different scales or resolutions [22-24]. The topological features are then displayed in persistence diagrams that show when each feature appears and disappears. These 2-dimensional diagrams can summarize large and high-dimensional data sets in order to perform machine learning algorithms for classification and clustering, but to do this one must properly define and compute distances on the space of persistence diagrams. Although there are various distances, the focus in this paper is the consideration of the popular Wasserstein distance, and the one defined in [15], which has been proven to be advantageous in machine learning tasks, e.g., see [25].