  This paper considers the noisy sparse phase retrieval problem: recovering a
sparse signal $x \in \mathbb{R}^p$ from noisy quadratic measurements $y_j =
(a_j' x )^2 + \epsilon_j$, $j=1, \ldots, m$, with independent sub-exponential
noise $\epsilon_j$. The goals are to understand the effect of the sparsity of
$x$ on the estimation precision and to construct a computationally feasible
estimator to achieve the optimal rates. Inspired by the Wirtinger Flow [12]
proposed for noiseless and non-sparse phase retrieval, a novel thresholded
gradient descent algorithm is proposed and it is shown to adaptively achieve
the minimax optimal rates of convergence over a wide range of sparsity levels
when the $a_j$'s are independent standard Gaussian random vectors, provided
that the sample size is sufficiently large compared to the sparsity of $x$.
