  Suppose that $\mathbf{y}=\lvert A\mathbf{x_0}\rvert+\eta$ where $\mathbf{x_0}
\in \mathbb{R}^d$ is the target signal and $\eta\in \mathbb{R}^m$ is a noise
vector. The aim of phase retrieval is to estimate $\mathbf{x_0}$ from
$\mathbf{y}$. A popular model for estimating $\mathbf{x_0} $ is the nonlinear
least square $ \widehat{\mathbf{x}}:={\rm argmin}_{\mathbf{x}} \| \lvert A
\mathbf{x}\rvert-\mathbf{y}\|_2$. One already develops many efficient
algorithms for solving the model, such as the seminal error reduction
algorithm. In this paper, we present the estimation performance of the model
with proving that $\|\widehat{\mathbf{x}}-\mathbf{x_0} \|\lesssim
{\|\eta\|_2}/{\sqrt{m}}$ under the assumption of $A$ being a Gaussian random
matrix. We also prove the reconstruction error ${\|\eta\|_2}/{\sqrt{m}}$ is
sharp. For the case where $\mathbf{x_0}$ is sparse, we study the estimation
performance of both the nonlinear Lasso of phase retrieval and its
unconstrained version. Our results are non-asymptotic, and we do not assume any
distribution on the noise $\eta$. To the best of our knowledge, our results
represent the first theoretical guarantee for the nonlinear least square and
for the nonlinear Lasso of phase retrieval.
