This paper introduces a novel framework for phase retrieval, called PhaseLift, which combines structured illumination with convex programming to recover the phase of a signal from intensity measurements. The method addresses the challenge of recovering a complex-valued signal from the magnitude of its Fourier transform, a problem that arises in various applications such as X-ray crystallography, diffraction imaging, and astronomical imaging. PhaseLift leverages the concept of matrix completion, transforming the problem into a rank minimization task, which can be efficiently solved using convex optimization techniques. The approach is robust to noise and can recover the phase of a signal from a small number of diffraction patterns. The paper demonstrates that three diffraction patterns can uniquely determine the phase of a signal under certain conditions. Theoretical analysis shows that the method is stable and can handle noisy data. Numerical experiments validate the effectiveness of PhaseLift in recovering signals from intensity measurements, showing high accuracy and robustness. The method is particularly useful for applications where the signal is sparse or has a known structure, and it provides a principled way to handle various noise models. The framework is flexible and can be adapted to different types of signals and measurement scenarios.This paper introduces a novel framework for phase retrieval, called PhaseLift, which combines structured illumination with convex programming to recover the phase of a signal from intensity measurements. The method addresses the challenge of recovering a complex-valued signal from the magnitude of its Fourier transform, a problem that arises in various applications such as X-ray crystallography, diffraction imaging, and astronomical imaging. PhaseLift leverages the concept of matrix completion, transforming the problem into a rank minimization task, which can be efficiently solved using convex optimization techniques. The approach is robust to noise and can recover the phase of a signal from a small number of diffraction patterns. The paper demonstrates that three diffraction patterns can uniquely determine the phase of a signal under certain conditions. Theoretical analysis shows that the method is stable and can handle noisy data. Numerical experiments validate the effectiveness of PhaseLift in recovering signals from intensity measurements, showing high accuracy and robustness. The method is particularly useful for applications where the signal is sparse or has a known structure, and it provides a principled way to handle various noise models. The framework is flexible and can be adapted to different types of signals and measurement scenarios.