  We present a $G_0W_0$ implementation that assesses the two major bottlenecks
of traditional plane-waves implementations, the summations over conduction
states and the inversion of the dielectric matrix, without introducing new
approximations in the formalism. The first bottleneck is circumvented by
converting the summations into Sternheimer equations. Then, the novel avenue of
expressing the dielectric matrix in a Lanczos basis is developed, which reduces
the matrix size by orders of magnitude while being computationally efficient.
We also develop a model dielectric operator that allows us to further reduce
the size of the dielectric matrix without accuracy loss. Furthermore, we
develop a scheme that reduces the numerical cost of the contour deformation
technique to the level of the lightest plasmon pole model. Finally, the use of
the simplified quasi-minimal residual scheme in replacement of the conjugate
gradients algorithm allows a direct evaluation of the $G_0W_0$ corrections at
the desired real frequencies, without need for analytical continuation. The
performance of the resulting $G_0W_0$ implementation is demonstrated by
comparison with a traditional plane-waves implementation, which reveals a
500-fold speedup for the silane molecule. Finally, the accuracy of our $G_0W_0$
implementation is demonstrated by comparison with other $G_0W_0$ calculations
and experimental results.
