  We establish that the initial value problem for the quadratic non-linear
Schr\"odinger equation $$ iu_t - \Delta u = u^2$$ where $u: \R^2 \times \R \to
\C$, is locally well-posed in $H^s(\R^2)$ when $s > -1$. The critical exponent
for this problem is $s_c=-1$ and previous work in \cite{c1} established local
well-posedness for $s > -3/4$.
