  We consider an $\ell_2$-regularized non-convex optimization problem for
recovering signals from their noisy phaseless observations. We design and study
the performance of a message passing algorithm that aims to solve this
optimization problem. We consider the asymptotic setting $m,n \rightarrow
\infty$, $m/n \rightarrow \delta$ and obtain sharp performance bounds, where
$m$ is the number of measurements and $n$ is the signal dimension. We show that
for complex signals the algorithm can perform accurate recovery with only
$m=\left ( \frac{64}{\pi^2}-4\right)n\approx 2.5n$ measurements. Also, we
provide sharp analysis on the sensitivity of the algorithm to noise. We
highlight the following facts about our message passing algorithm: (i) Adding
$\ell_2$ regularization to the non-convex loss function can be beneficial even
in the noiseless setting; (ii) spectral initialization has marginal impact on
the performance of the algorithm.
