  In recent years, there has been a surge of interest in proving discretization
bounds for diffusion models. These works show that for essentially any data
distribution, one can approximately sample in polynomial time given a
sufficiently accurate estimate of its score functions at different noise
levels. In this work, we propose a new discretization scheme for diffusion
models inspired by Shen and Lee's randomized midpoint method for log-concave
sampling~\cite{ShenL19}. We prove that this approach achieves the best known
dimension dependence for sampling from arbitrary smooth distributions in total
variation distance ($\widetilde O(d^{5/12})$ compared to $\widetilde
O(\sqrt{d})$ from prior work). We also show that our algorithm can be
parallelized to run in only $\widetilde O(\log^2 d)$ parallel rounds,
constituting the first provable guarantees for parallel sampling with diffusion
models.
  As a byproduct of our methods, for the well-studied problem of log-concave
sampling in total variation distance, we give an algorithm and simple analysis
achieving dimension dependence $\widetilde O(d^{5/12})$ compared to $\widetilde
O(\sqrt{d})$ from prior work.
