  We prove that the Cauchy problem for the 2D quintic defocusing biharmonic
Schr\"odinger equation is globally well-posed in the Sobolev spaces
$H^s(\mathbb{R}^2)$ for $\frac{8}{7}<s<2$. Our main ingredient to establish the
result is the $I$-method of Colliander-Keel-Staffilani-Takaoka-Tao
\cite{colliander2002almost} which is used to construct the modified energy
functional that is almost conserved in time.
