  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.
