This paper introduces a novel framework called PhaseLift for phase retrieval, a problem common in X-ray crystallography, diffraction imaging, and astronomical imaging. PhaseLift combines multiple structured illuminations with convex programming to recover the phase from intensity measurements, typically the modulus of the diffracted wave. The authors demonstrate that any complex-valued object can be recovered from a few diffracted patterns by solving a simple convex optimization problem inspired by matrix completion. They also show that their noise-aware algorithms are stable, degrading gracefully as the signal-to-noise ratio decreases. The paper includes theoretical results showing that three diffracted patterns can uniquely determine the phase of an object under certain conditions. The methodology involves lifting the phase retrieval problem into a rank-one matrix completion problem and formulating it as a convex program. The authors provide numerical experiments to illustrate the effectiveness of PhaseLift, including both noiseless and noisy data scenarios.This paper introduces a novel framework called PhaseLift for phase retrieval, a problem common in X-ray crystallography, diffraction imaging, and astronomical imaging. PhaseLift combines multiple structured illuminations with convex programming to recover the phase from intensity measurements, typically the modulus of the diffracted wave. The authors demonstrate that any complex-valued object can be recovered from a few diffracted patterns by solving a simple convex optimization problem inspired by matrix completion. They also show that their noise-aware algorithms are stable, degrading gracefully as the signal-to-noise ratio decreases. The paper includes theoretical results showing that three diffracted patterns can uniquely determine the phase of an object under certain conditions. The methodology involves lifting the phase retrieval problem into a rank-one matrix completion problem and formulating it as a convex program. The authors provide numerical experiments to illustrate the effectiveness of PhaseLift, including both noiseless and noisy data scenarios.