The Score Hamiltonian: Mapping Diffusion Models to Adiabatic Transport
For the machine learning community, this work provides a theoretical foundation linking diffusion models to quantum adiabatic transport, offering new insights into sampling efficiency and error bounds.
The paper establishes an exact correspondence between score-based diffusion models and adiabatic transport in quantum mechanics, deriving novel density reconstruction bounds and principled annealing schedules. It identifies the fundamental sampling limit as the ratio of squared score-matching error to the spectral gap of the Score Hamiltonian.
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.