The Score Hamiltonian: Mapping Diffusion Models to Adiabatic Transport
We exhibit an exact correspondence between sampling with score-based diffusion models and adiabatic transport of ground states for a family of Schrödinger 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 Poincaré constant of the data density.
| Authors | Peter Halmos, Boris Hanin |
|---|---|
| Primary category | math-ph |
| Published | 28 May 2026 |
| Last updated at source | 28 July 2026 |
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