Quantum spectroscopy of topological dynamics via a supersymmetric Hamiltonian

TL;DR

The work introduces a quantum spectroscopic framework that connects time-domain dynamics to the spectrum of a supersymmetric (SUSY) Hamiltonian, enabling direct reading of topological invariants from quantum hardware. By mapping a data-derived simplicial complex to the SUSY Laplacian, zero modes yield Betti numbers and the first nonzero eigenvalue tracks the stability of topological features, linking persistence to measurable energy gaps. Using Takens embedding of the Lorenz attractor and a resource-efficient single-ancilla QPE on IBM hardware, the authors demonstrate that the SUSY gap $\\ abla^{(1)}_{\\mathrm{SUSY}}$ co-varies with the classical $H_1$ persistence, with a two-stage transition: onset of chaos followed by topological maturation around $\\rho \\approx 41$. The results suggest a practical quantum topology spectrometer capable of probing data geometry beyond classical diagonalization limits, with potential extensions to higher homology and fully quantum filtrations, offering a path toward quantum-accelerated TDA in complex dynamical systems.

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.

Figures (15)

• Figure 1: Quantum spectroscopy pipeline for topological–dynamical analysis. The workflow converts a scalar time series into a quantum-mechanical spectrum whose low-lying energies encode topological structure. A Lorenz signal is Takens-embedded, topology-aware point selection preserves loop geometry, and a supersymmetric (SUSY) Hamiltonian is constructed and simulated by a controlled time-evolution circuit. Single-ancilla quantum phase estimation (QPE) retrieves the eigenvalue spectrum, where near-zero modes correspond to harmonic $H_1$ loops.
• Figure 2: Five-point validation of quantum Betti estimation. Three Vietoris–Rips filtrations of a five-point complex demonstrate how quantum spectra track topological change. (A) At $\varepsilon=0.8$, a single loop yields $\beta_1=1$; (B) at $\varepsilon=0.9$, one triangle forms but the loop persists; (C) at $\varepsilon=1.0$, the complex becomes contractible ($\beta_1=0$). QPE spectra reproduce the classical Hodge–Laplacian eigenvalues, confirming accurate quantum detection of loop annihilation.
• Figure 3: Dynamical and topological diagnostics across the Lorenz parameter $\rho$. (A) Spectral entropy, (B) free-energy curvature, (C) ground-state fidelity, (D) maximum Lyapunov exponent, (E) $H_1$ persistence, and (F) low-energy gap $\gamma(\rho)=E_1-E_0$. The joint evolution of these indicators suggests two transitions: a chaotic onset near $\rho\approx30$ and topological stabilization near $\rho\approx41$.
• Figure 4: Quantum–classical correspondence of topological stabilization. Comparison of the Hodge–Laplacian spectrum energy gap $\Delta^{(1)}_{\mathrm{SUSY}}$ (blue/orange bars: simulator vs IBM hardware) with the highest classical $H_1$ persistence $\ell^{\max}_{H_1}$ (purple line) as functions of the Rayleigh parameter $\rho$. Both peak near $\rho\approx41$, where the Lorenz attractor’s loop geometry and the quantum harmonic subspace show their strongest alignment in this study, suggesting a possible spectral–topological correspondence. Agreement (IBM vs simulation): Pearson $r=0.897$ ($p=0.0062$), Spearman $\rho=0.815$ ($p=0.025$), bias $-0.042$, MAE $0.042$, RMSE $0.091$, MAPE $9.9\%$, $R^2=0.701$. Link to classical topology:$r(H_1,\mathrm{Sim})=0.798$, $r(H_1,\mathrm{IBM})=0.606$.
• Figure S1: Controlled time evolution for one local term $U_\ell(t)=e^{-i\theta_\ell t P_{q_1}\!\otimes P_{q_2}\!\otimes P_{q_3}}$. Basis rotations convert $X/Y$ to $Z$, CNOTs collect parity on $q_3$, and an ancilla-controlled $R_Z\!\left(2\theta_\ell t\right)$ realizes the exponential.