Quantum spectroscopy of topological dynamics via a supersymmetric Hamiltonian

TL;DR

This paper introduces a quantum spectroscopy method using a supersymmetric Hamiltonian to analyze topological features of complex dynamical systems, demonstrating advantages over classical approaches and insights into chaos and attractor geometry.

Contribution

It reinterprets topological data analysis as quantum spectral estimation via a SUSY Hamiltonian, enabling efficient analysis of high-dimensional data.

Findings

Quantum spectroscopy estimates topological invariants.

Spectral gap tracks persistence of topological features.

Reopening of the gap indicates geometric maturation.

Abstract

Topological data analysis (TDA) characterizes 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 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. Validated on small complexes yet offering an exponential advantage over classical diagonalization (from $O(N^3)$ to $\mathrm{poly}(\log N)$), this framework suggests that quantum hardware can function as a spectrometer for data topologies beyond classical reach.