We prove that the Cauchy problem for the 2D quintic defocusing biharmonic
Schr\"odinger equation is globally well-posed in the Sobolev spaces
Hs(R2) for 78<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 conserved in time