  The paper aims to study the performance of the amplitude-based model \newline
$\widehat{\mathbf x} \in {\rm argmin}_{{\mathbf x}\in
\mathbb{C}^d}\sum_{j=1}^m\left(|\langle {\mathbf a}_j,{\mathbf
x}\rangle|-b_j\right)^2$, where $b_j:=|\langle {\mathbf a}_j,{\mathbf
x}_0\rangle|+\eta_j$ and ${\mathbf x}_0\in \mathbb{C}^d$ is a target signal.
The model is raised in phase retrieval as well as in absolute value
rectification neural networks. Many efficient algorithms have been developed to
solve it in the past decades. {However, there are very few results available
regarding the estimation performance in the complex case under noisy
conditions.} In this paper, {we present a theoretical guarantee on the
amplitude-based model for the noisy complex phase retrieval problem}.
Specifically, we show that $\min_{\theta\in[0,2\pi)}\|\widehat{\mathbf
x}-\exp(\mathrm{i}\theta)\cdot{\mathbf x}_0\|_2 \lesssim \frac{\|{\mathbf
\eta}\|_2}{\sqrt{m}}$ holds with high probability provided the measurement
vectors ${\mathbf a}_j\in \mathbb{C}^d,$ $j=1,\ldots,m,$ are {i.i.d.} complex
sub-Gaussian random vectors and $m\gtrsim d$. Here ${\mathbf
\eta}=(\eta_1,\ldots,\eta_m)\in \mathbb{R}^m$ is the noise vector without any
assumption on the distribution. Furthermore, we prove that the reconstruction
error is sharp. For the case where the target signal ${\mathbf x}_0\in
\mathbb{C}^{d}$ is sparse, we establish a similar result for the nonlinear
constrained $\ell_1$ minimization model. { To accomplish this, we leverage a
strong version of restricted isometry property for an operator on the space of
simultaneous low-rank and sparse matrices.}
