  We consider the Cauchy problem for the cubic fourth order nonlinear
Schr\"odinger equation (4NLS) on the circle. In particular, we prove global
well-posedness of the renormalized 4NLS in negative Sobolev spaces
$H^s(\mathbb{T})$, $s > -\frac{1}{3}$, with enhanced uniqueness. The proof
consists of two separate arguments. (i) We first prove global existence in
$H^s(\mathbb{T})$, $s > -\frac{9}{20}$, via the short-time Fourier restriction
norm method. By following the argument in Guo-Oh for the cubic NLS, this also
leads to non-existence of solutions for the (non-renormalized) 4NLS in negative
Sobolev spaces. (ii) We then prove enhanced uniqueness in $H^s(\mathbb{T})$, $s
> -\frac{1}{3}$, by establishing an energy estimate for the difference of two
solutions with the same initial condition. For this purpose, we perform an
infinite iteration of normal form reductions on the $H^s$-energy functional,
allowing us to introduce an infinite sequence of correction terms to the
$H^s$-energy functional in the spirit of the $I$-method. In fact, the main
novelty of this paper is this reduction of the $H^s$-energy functionals (for a
single solution and for the difference of two solutions with the same initial
condition) to sums of infinite series of multilinear terms of increasing
degrees.
