  The Bogolyubov-Ruzsa lemma, in particular the quantitative bounds obtained by
Sanders, plays a central role in obtaining effective bounds for the inverse
$U^3$ theorem for the Gowers norms. Recently, Gowers and Mili\'cevi\'c applied
a bilinear Bogolyubov-Ruzsa lemma as part of a proof of the inverse $U^4$
theorem with effective bounds. The goal of this note is to obtain quantitative
bounds for the bilinear Bogolyubov-Ruzsa lemma which are similar to those
obtained by Sanders for the Bogolyubov-Ruzsa lemma.
  We show that if a set $A \subset \mathbb{F}_p^n \times \mathbb{F}_p^n$ has
density $\alpha$, then after a constant number of horizontal and vertical sums,
the set $A$ would contain a bilinear structure of co-dimension $r=\log^{O(1)}
\alpha^{-1}$. This improves the results of Gowers and Mili\'cevi\'c which
obtained similar results with a weaker bound of $r=\exp(\exp(\log^{O(1)}
\alpha^{-1}))$ and by Bienvenu and L\^e which obtained
$r=\exp(\exp(\exp(\log^{O(1)} \alpha^{-1})))$.
