Quantum topological data analysis: using Fourier analysis to learn topological properties*

The topological features of datasets find application in several industrial sectors where explainable AI is crucial, including medical image analysis, transportation optimization, molecular structure analysis in chemistry, and explainable features in financial analysis. Quantum topological data analysis (QTDA) aims to harness quantum systems to find efficient ways to estimate these features. In this work, we develop quantum protocols for topological data analysis on NISQ platforms and beyond. Using tools typical of many-body physics, quantum thermodynamics, and quantum machine learning (QML), we present novel approaches to the problem of Betti number estimation. In this talk, we will show these algorithms, leveraging Fourier-like transforms and Cartan decomposition. We extract topological features by analyzing the combinatorial Laplacian dynamics and its underlying spectral properties. Additionally, we build models based on quantum convolutional neural networks (QCNN) for learning and predicting properties of simplicial complexes. Finally, to back the industrial relevance of our findings, we will show example runs onto a selection of currently available quantum processing units (QPUs).