Quantum spectroscopy of topological dynamics via a supersymmetric Hamiltonian

Topological data analysis (TDA) involves complex dynamics through global invariants, but classical computation becomes prohibitive for high-dimensional data. We reinterpret time-domain dynamics as the eigenvalue spectrum of a supersymmetric (SUSY) Hamiltonian and thereby estimate topological descriptors through quantum spectroscopy. While zero modes correspond to Betti numbers, we show that low-lying excited states quantify the stability of topological features. Using a Takens embedding of the Lorenz system together with a resource-efficient 1 quantum phase estimation implemented on IBM quantum hardware, we observe that the spectral gap of the SUSY Laplacian tracks the persistence of homological structures. Notably, the minimum of this spectral gap coincides with the onset of chaos, whereas its reopening reflects the geometric maturation of the attractor. This framework is validated on small complexes and suggests a potential asymp-totic scaling benefit over classical diagonalization. This framework indicates that quantum hardware can function as a spectrometer for data topologies beyond classical reach.