  A method for analyzing non-asymptotic guarantees of numerical discretizations
of ergodic SDEs in Wasserstein-2 distance is presented by Sanz-Serna and
Zygalakis in ``Wasserstein distance estimates for the distributions of
numerical approximations to ergodic stochastic differential equations". They
analyze the UBU integrator which is strong order two and only requires one
gradient evaluation per step, resulting in desirable non-asymptotic guarantees,
in particular $\mathcal{O}(d^{1/4}\epsilon^{-1/2})$ steps to reach a distance
of $\epsilon > 0$ in Wasserstein-2 distance away from the target distribution.
However, there is a mistake in the local error estimates in Sanz-Serna and
Zygalakis (2021), in particular, a stronger assumption is needed to achieve
these complexity estimates. This note reconciles the theory with the dimension
dependence observed in practice in many applications of interest.
