Quantum topological data analysis via the estimation of the density of states

We develop a quantum topological data analysis (QTDA) protocol based on the estimation of the density of states (DOS) of the combinatorial Laplacian. Computing topological features of graphs and simplicial complexes is crucial for analyzing datasets and building explainable AI solutions. This task becomes computationally hard for simplicial complexes with over sixty vertices and high-degree topological features due to a combinatorial scaling. We propose to approach the task by embedding underlying hypergraphs as effective quantum Hamiltonians and evaluating their density of states from the time evolution. Specifically, we compose propagators as quantum circuits using the Cartan decomposition of effective Hamiltonians and sample overlaps of time-evolved states using multi-fidelity protocols. Next, we develop various post-processing routines and implement a Fourier-like transform to recover the rank (and kernel) of Hamiltonians. This enables us to estimate the Betti numbers, revealing the topological features of simplicial complexes. We test our protocol on noiseless and noisy quantum simulators and run examples on IBM quantum processors. We observe the resilience of the proposed QTDA approach to real-hardware noise even in the absence of error mitigation, showing the promise to near-term device implementations and highlighting the utility of global DOS-based estimators.