  In this article we consider the smoothing problem for hidden Markov models
(HMM). Given a hidden Markov chain $\{X_n\}_{n\geq 0}$ and observations
$\{Y_n\}_{n\geq 0}$, our objective is to compute
$\mathbb{E}[\varphi(X_0,\dots,X_k)|y_{0},\dots,y_n]$ for some real-valued,
integrable functional $\varphi$ and $k$ fixed, $k \ll n$ and for some
realisation $(y_0,\dots,y_n)$ of $(Y_0,\dots,Y_n)$. We introduce a novel
application of the multilevel Monte Carlo (MLMC) method with a coupling based
on the Knothe-Rosenblatt rearrangement. We prove that this method can
approximate the afore-mentioned quantity with a mean square error (MSE) of
$\mathcal{O}(\epsilon^2)$, for arbitrary $\epsilon>0$ with a cost of
$\mathcal{O}(\epsilon^{-2})$. This is in contrast to the same direct Monte
Carlo method, which requires a cost of $\mathcal{O}(n\epsilon^{-2})$ for the
same MSE. The approach we suggest is, in general, not possible to implement, so
the optimal transport methodology of \cite{span} is used, which directly
approximates our strategy. We show that our theoretical improvements are
achieved, even under approximation, in several numerical examples.
