  This paper considers the question of recovering the phase of an object from
intensity-only measurements, a problem which naturally appears in X-ray
crystallography and related disciplines. We study a physically realistic setup
where one can modulate the signal of interest and then collect the intensity of
its diffraction pattern, each modulation thereby producing a sort of coded
diffraction pattern. We show that PhaseLift, a recent convex programming
technique, recovers the phase information exactly from a number of random
modulations, which is polylogarithmic in the number of unknowns. Numerical
experiments with noiseless and noisy data complement our theoretical analysis
and illustrate our approach.
