Clustering using Quantum Persistent Homology

Abstract

Persistent homology is a way to analyse the shape of a dataset rather than its semantics. This is done by assigning the dataset holes in different dimensions, borrowing this theory from the mathematical area of topology. The dataset is treated as being sampled from a topological space, which is in turn approximated at different resolutions. The persistent Betti numbers then count the number of holes in these spaces, registering for how long the holes persist as the resolution increases. Computing such features classically may prove costly, with some steps possibly requiring exponential time. In a 2016 paper, a polynomial time quantum algorithm for the approximation of persistent homology is promised. In this thesis it is shown these methods are generally incapable of uniquely determining the persistent Betti numbers in arbitrary dimensions. The algorithm does succeed in the zeroth dimension. After treating the prerequisites, the methods from this paper are explained, the above statements are proven, and finally the quantum algorithm is related to agglomerative hierarchical clustering and spectral clustering.