  We show that any $n$-variate polynomial computable by a syntactically
multilinear circuit of size $\operatorname{poly}(n)$ can be computed by a
depth-$4$ syntactically multilinear ($\Sigma\Pi\Sigma\Pi$) circuit of size at
most $\exp\left({O\left(\sqrt{n\log n}\right)}\right)$. For degree $d =
\omega(n/\log n)$, this improves upon the upper bound of
$\exp\left({O(\sqrt{d}\log n)}\right)$ obtained by Tavenas~\cite{T15} for
general circuits, and is known to be asymptotically optimal in the exponent
when $d < n^{\epsilon}$ for a small enough constant $\epsilon$. Our upper bound
matches the lower bound of $\exp\left({\Omega\left(\sqrt{n\log
n}\right)}\right)$ proved by Raz and Yehudayoff~\cite{RY09}, and thus cannot be
improved further in the exponent. Our results hold over all fields and also
generalize to circuits of small individual degree.
  More generally, we show that an $n$-variate polynomial computable by a
syntactically multilinear circuit of size $\operatorname{poly}(n)$ can be
computed by a syntactically multilinear circuit of product-depth $\Delta$ of
size at most $\exp\left(O\left(\Delta \cdot (n/\log n)^{1/\Delta} \cdot \log
n\right)\right)$. It follows from the lower bounds of Raz and Yehudayoff (CC
2009) that in general, for constant $\Delta$, the exponent in this upper bound
is tight and cannot be improved to $o\left(\left(n/\log
n\right)^{1/\Delta}\cdot \log n\right)$.
