  We show how to sample in parallel from a distribution $\pi$ over $\mathbb
R^d$ that satisfies a log-Sobolev inequality and has a smooth log-density, by
parallelizing the Langevin (resp. underdamped Langevin) algorithms. We show
that our algorithm outputs samples from a distribution $\hat\pi$ that is close
to $\pi$ in Kullback--Leibler (KL) divergence (resp. total variation (TV)
distance), while using only $\log(d)^{O(1)}$ parallel rounds and
$\widetilde{O}(d)$ (resp. $\widetilde O(\sqrt d)$) gradient evaluations in
total. This constitutes the first parallel sampling algorithms with TV distance
guarantees.
  For our main application, we show how to combine the TV distance guarantees
of our algorithms with prior works and obtain RNC sampling-to-counting
reductions for families of discrete distribution on the hypercube $\{\pm 1\}^n$
that are closed under exponential tilts and have bounded covariance.
Consequently, we obtain an RNC sampler for directed Eulerian tours and
asymmetric determinantal point processes, resolving open questions raised in
prior works.
