Diffusion Models as Stochastic Quantization in Lattice Field Theory

TL;DR

The paper uncovers a direct link between diffusion models (DMs) and stochastic quantization (SQ) in lattice field theory, and demonstrates that a DM can serve as a global sampler to generate lattice configurations for a two-dimensional $\phi^4$ theory. By training a score-based network to learn the reverse diffusion, the authors show that denoising from noise can reproduce the target distribution and reduce autocorrelation times, especially near criticality where traditional MCMC suffers slowing down. They frame the DM dynamics in SQ terms, with an effective action $S_{\rm DM}$ and a probability-flow formulation that allows likelihood-based assessment via the Skilling-Hutchinson estimator. The results indicate that DMs can accurately capture physical observables (e.g., magnetization, susceptibility, Binder cumulant) and significantly accelerate sampling, offering a path toward efficient ensemble generation in lattice theories and potential extensions to gauge fields and complex weights. Overall, this work provides a novel, executable bridge between modern diffusion-based generative modeling and established stochastic quantization techniques, with practical implications for speeding up lattice simulations and guiding future explorations in more challenging theories.

Abstract

In this work, we establish a direct connection between generative diffusion models (DMs) and stochastic quantization (SQ). The DM is realized by approximating the reversal of a stochastic process dictated by the Langevin equation, generating samples from a prior distribution to effectively mimic the target distribution. Using numerical simulations, we demonstrate that the DM can serve as a global sampler for generating quantum lattice field configurations in two-dimensional $φ^4$ theory. We demonstrate that DMs can notably reduce autocorrelation times in the Markov chain, especially in the critical region where standard Markov Chain Monte-Carlo (MCMC) algorithms experience critical slowing down. The findings can potentially inspire further advancements in lattice field theory simulations, in particular in cases where it is expensive to generate large ensembles.

Figures (15)

• Figure 1: Sketch of a Langevin simulation. Each line represents a different trajectory, evolving from an initial state at $\tau=0$ to one at a later time, $T$. The initial configurations are sampled from a prior but naive distribution, $\phi\sim P[\phi,\tau=0]$. After the simulation, the final configurations follow the equilibrated distribution, $\phi\sim P[\phi,\tau=T]\sim P_{\rm target}[\phi]$.
• Figure 2: A sketch of the forward diffusion process (left panel) and the reverse denoising process (right panel). The two stochastic processes are described by two stochastic differential equations. The target distribution is typically unknown but learnt from the training data.
• Figure 3: Drift terms (upper row) and effective actions (middle row) learned by the DM as a function of $\phi$ in both single-well (left column) and double-well (right column) actions, for various values of the time $\tau$ during the stochastic process. The action is shifted by a constant $\Delta S_0$. The dashed lines indicate the exact values. 1024 samples generated using the exact and the DM are shown in the bottom row.
• Figure 4: The denoising process for generating four independent configurations from a well-trained DM.
• Figure 5: Comparison of the distribution of the magnetization with 1024 samples generated from the well-trained DM and using HMC.