Lower Bounds for Parallel Diffusion Sampling

Standard diffusion samplers generate samples through repeated evaluations of a learned score function. Parallel sampling methods seek to accelerate generation by trading additional evaluations for fewer sequential rounds. This raises the question of how much sequential dependence is unavoidable, even when many score queries can be made simultaneously. We establish the first polynomial parallel-round lower bounds for diffusion sampling with approximate scores. Specifically, we prove (1) a $\widetildeΩ(d^{1/3})$-round lower bound for sampling smooth, near-isotropic Gaussian mixtures in $R^d$, and (2) an $Ω(d)$-round lower bound for uniform sampling from anisotropic axis-aligned boxes contained in the unit ball. Both bounds hold for arbitrary randomized algorithms making polynomially many queries per round at arbitrary locations and noise levels, with inverse-polynomial score error and constant total variation accuracy. The linear bound is tight for our box family. Our constructions use fixed approximate score oracles that enforce sequential access to hidden information while satisfying the accuracy guarantee at every noise level.