The Score Hamiltonian: Mapping Diffusion Models to Adiabatic Transport

2026-05-28

AI Summary

This work establishes a fundamental connection between diffusion model sampling and adiabatic transport in quantum mechanics. By introducing a novel “Score Hamiltonian,” the authors map the score-based diffusion process to the adiabatic evolution of the ground state of a Schrödinger operator. Leveraging an adiabatic theorem for the time-dependent Fokker–Planck equation, they rigorously formalize this correspondence for the first time. Key contributions include the development of the Score Hamiltonian framework, derivation of a fundamental sampling accuracy limit governed jointly by the spectral gap and score-matching error, and an upper bound on density reconstruction error. These theoretical insights lead to a principled, theory-driven annealing schedule. The analysis reveals that sampling performance is ultimately constrained by the ratio of the squared score error to the inverse Poincaré constant of the data distribution.

Abstract

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.

Problem

• score-based diffusion models
• adiabatic transport
• Schrödinger operators
• sampling limits
• spectral gap

Innovation

• Score Hamiltonian
• adiabatic transport
• diffusion models
• spectral gap
• Fokker-Planck equation