A topological features based quantum kernel

Topological data analysis (TDA) has emerged as an effective technique to derive valuable insights from intricate data. In TDA we typically embed our data into simplicial complexes from which we can extract useful geometrical-invariant descriptors such as the Betti numbers. These techniques can be employed to define kernels that can be incorporated within existing machine learning algorithms. These kernels have widespread utility, as they are built upon robust mathematical frameworks that offer theoretical assurances regarding their effectiveness. However, the computation of higher-dimensional Betti numbers can be prohibitively expensive on classical hardware, while quantum algorithms can approximate them with some speed-up. In this work, we propose a quantum approach to defining topological kernels, which is based on constructing Betti curves, i.e. topological fingerprint of filtrations with increasing order. By integrating with any kernelized methods, this approach holds the potential to provide an advantage in addressing data-intensive machine learning tasks.