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arXiv · 2606.05217

The Score Hamiltonian: Mapping Diffusion Models to Adiabatic Transport

Peter Halmos·Boris Hanin
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Abstract

We exhibit an exact correspondence between sampling with score-based diffusion models and adiabatic transport of ground states for a family of Schr\"odinger operators we call Score Hamiltonians, built from the learned score's quantum potential. We obtain novel density reconstruction bounds and principled annealing schedules via adiabatic theorems for Fokker-Planck equations with time-varying potentials. We find the fundamental limit of sampling is set by the ratio of squared score-matching error to Score Hamiltonian spectral gap - the inverse Poincar\'e constant of the data density.

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Peter Halmos, Boris Hanin. 2026-05-28. The Score Hamiltonian: Mapping Diffusion Models to Adiabatic Transport. https://arxiv.org/abs/2606.05217

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