  The $GW$ method is widely used for calculating the electronic band structure
of materials. The high computational cost of $GW$ algorithms prohibits their
application to many systems of interest. We present a periodic, low-scaling and
highly efficient $GW$ algorithm that benefits from the locality of the Gaussian
basis and the polarizability. The algorithm enables $G_0W_0$ calculations on a
MoSe$_2$/WS$_2$ bilayer with 984 atoms per unit cell, in 42 hours using 1536
cores. This is four orders of magnitude faster than a plane-wave $G_0W_0$
algorithm, allowing for unprecedented computational studies of electronic
excitations at the nanoscale.
