  Topological data analysis (TDA) has emerged as a powerful tool for extracting
meaningful insights from complex data. TDA enhances the analysis of objects by
embedding them into a simplicial complex and extracting useful global
properties such as the Betti numbers, i.e. the number of multidimensional
holes, which can be used to define kernel methods that are easily integrated
with existing machine-learning algorithms. These kernel methods have found
broad applications, as they rely on powerful mathematical frameworks which
provide theoretical guarantees on their performance. However, the computation
of higher-dimensional Betti numbers can be prohibitively expensive on classical
hardware, while quantum algorithms can approximate them in polynomial time in
the instance size. 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. We exhibit a
working prototype of our approach implemented on a noiseless simulator and show
its robustness by means of some empirical results suggesting that topological
approaches may offer an advantage in quantum machine learning.
