This paper considers the noisy sparse phase retrieval problem: recovering a sparse signal x ∈ R p from noisy quadratic measurements y j = (a j x) 2 + ε j , j = 1, . . . , m, with independent sub-exponential noise ε 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 adaptively. Inspired by the Wirtinger Flow [IEEE Trans. Inform. Theory 61 (2015) 1985-2007 proposed for

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... arse and noiseless 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.