  We consider the Cauchy problem for the fourth order cubic nonlinear
Schr\"odinger equation (4NLS). The main goal of this paper is to prove low
regularity well-posedness and mild ill-posedness for (4NLS). We prove three
results. First, we show that (4NLS) is locally well-posed in
$H^s\left(\mathbb{R}\right), s\geq -\frac{1}{2}$ using the Fourier restriction
norm method. Second, we show that (4NLS) is globally well-posed in
$H^s\left(\mathbb{R}\right),s\geq -\frac{1}{2}$. To prove this, we use the
$I$-method with the correction term strategy presented in
Colliander-Keel-Staffilani-Takaoka-Tao [7]. Finally, we prove that (4NLS) is
mildly ill-posed in the sense that the flow map fails to be locally uniformly
continuous in $H^s(\mathbb{R}), s<-\frac{1}{2}$. Therefore, these results show
that $s=-\frac{1}{2}$ is the sharp regularity threshold for which the
well-posedness problem can be dealt with an iteration argument.
