  We investigate the properties of the semiclassical short periodic orbit
approach for the study of open quantum maps that was recently introduced in [M.
Novaes, J.M. Pedrosa, D. Wisniacki, G.G. Carlo, and J.P. Keating, Phys. Rev. E
80, 035202(R) 2009]. We provide conclusive numerical evidence, for the
paradigmatic systems of the open baker and cat maps, that by using this
approach the dimensionality of the eigenvalue problem is reduced according to
the fractal Weyl law. The method also reproduces the projectors
$|\psi^R_n><\psi_n^L|$, which involves the right and left states associated
with a given eigenvalue and is supported on the classical phase space repeller.
