  In this paper, we consider the KPZ equation driven by space-time white noise
replaced with its fractional derivatives of order $\gamma>0$ in spatial
variable. A well-posedness theory for the KPZ equation is established by Hairer
[3] as an application of the theory of regularity structures. Our aim is to see
to what extent his theory works if noises become rougher. We can expect that
his theory works if and only if $\gamma<1/2$. However, we show that the
renormalization like "$(\partial_x h)^2-\infty$" is well-posed only if
$\gamma<1/4$.
