The Score Hamiltonian: Mapping Diffusion Models to Adiabatic Transport
On the map: Diffusion models → Diffusion foundations and sampling
Abstract (arXiv)
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.
Key facts. No citing papers in the map yet (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 14 of its 47 references that are in the map. First posted on arXiv 28 May 2026.
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14 of 47 references.
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- 2025Are We Really Learning the Score Function? Reinterpreting Diffusion Models Through Wasserstein Gradient Flow Matching30 Aug
- 2023Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization7 Aug
- 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions22 Sep
- 2022Convergence for score-based generative modeling with polynomial complexity13 Jun
- 2021High-Resolution Image Synthesis with Latent Diffusion Models20 Dec
- 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
- 2020Denoising Diffusion Probabilistic Models19 Jun
Other references · 7
- 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
- 2015Variational Inference with Normalizing Flowscited by 2,170
- 2016Density estimation using Real NVPcited by 2,042
- 2022Equivariant Diffusion for Molecule Generation in 3Dcited by 677
- 2023Geometric Latent Diffusion Models for 3D Molecule Generationcited by 186
- 2024Statistical optimal transportcited by 47
- 2025Consistent Sampling and Simulation: Molecular Dynamics with Energy-Based Diffusion Modelscited by 23
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- 2023A Score-Based Model for Learning Neural Wavefunctions
- 2024Quantum State Generation with Structure-Preserving Diffusion Model
- 2025Mitigating Barren Plateaus in Quantum Denoising Diffusion Probabilistic Model
- 2026Scalable quantum simulation of continuous-time generative models via tensor networks
- 2026Quantum Circuits in Diffusion Models: A Fair-Comparison Study and a Mechanistic Analysis of Angle-Embedding Failures