Homology Computation of Large Point Clouds using Quantum Annealing

Homology is a tool in topological data analysis which measures the shape of the data. In many cases, these measurements translate into new insights which are not readily available by other means. To compute homology, we rely on mathematical constructions which scale exponentially with the size of the data. Therefore, for large point clouds, the computation is infeasible using classical computers. In this paper, we present a quantum annealing pipeline for computation of homology of large point clouds. It is designed to work concurrently with resizable cloud computing platforms. The pipeline takes as input a witness graph approximating the given point cloud. It uses quantum annealing to compute a clique covering of the graph and then uses this cover to construct a Mayer-Vietoris complex. The pipeline terminates by performing a simplified homology computation of the Mayer-Vietoris complex in parallel. We have designed three different clique coverings and their quantum annealing formulation with which our algorithm exhibits an exponential speed-up over classical implementations. In fact, not only the computation is simplified but also the simplicial complex construction itself is greatly simplified. We have also included tests using D-Wave 2X quantum processor.