  We study the stable recovery of complex $k$-sparse signals from as few
phaseless measurements as possible. The main result is to show that one can
employ $\ell_1$ minimization to stably recover complex $k$-sparse signals from
$m\geq O(k\log (n/k))$ complex Gaussian random quadratic measurements with high
probability. To do that, we establish that Gaussian random measurements satisfy
the restricted isometry property over rank-$2$ and sparse matrices with high
probability. This paper presents the first theoretical estimation of the
measurement number for stably recovering complex sparse signals from complex
Gaussian quadratic measurements.
