  We establish an information complexity lower bound of randomized algorithms
for simulating underdamped Langevin dynamics. More specifically, we prove that
the worst $L^2$ strong error is of order $\Omega(\sqrt{d}\, N^{-3/2})$, for
solving a family of $d$-dimensional underdamped Langevin dynamics, by any
randomized algorithm with only $N$ queries to $\nabla U$, the driving Brownian
motion and its weighted integration, respectively. The lower bound we establish
matches the upper bound for the randomized midpoint method recently proposed by
Shen and Lee [NIPS 2019], in terms of both parameters $N$ and $d$.
