  We consider the problem of recovering signals from their power spectral
density. This is a classical problem referred to in literature as the phase
retrieval problem, and is of paramount importance in many fields of applied
sciences. In general, additional prior information about the signal is required
to guarantee unique recovery as the mapping from signals to power spectral
density is not one-to-one. In this paper, we assume that the underlying signals
are sparse. Recently, semidefinite programming (SDP) based approaches were
explored by various researchers. Simulations of these algorithms strongly
suggest that signals upto $o(\sqrt{n})$ sparsity can be recovered by this
technique. In this work, we develop a tractable algorithm based on reweighted
$l_1$-minimization that recovers a sparse signal from its power spectral
density for significantly higher sparsities, which is unprecedented.
  We discuss the square-root bottleneck of the existing convex algorithms and
show that a $k$-sparse signal can be efficiently recovered using $O(k^2logn)$
phaseless Fourier measurements. We also show that a $k$-sparse signal can be
recovered using only $O(k log n)$ phaseless measurements if we are allowed to
design the measurement matrices.
