PhaseLift is a method for exactly and stably recovering signals from magnitude measurements using convex programming. The paper proves that if vectors $ z_i $ are sampled independently and uniformly at random on the unit sphere, a signal $ x \in \mathbb{C}^n $ can be recovered exactly (up to a global phase factor) by solving a semidefinite program—a trace-norm minimization problem. This holds with high probability when the number of measurements $ m $ is on the order of $ n \log n $. The method is robust to additive noise, and the recovery error is small when the data are corrupted by noise. The paper also shows that the combinatorial phase retrieval problem can be solved by convex programming techniques under certain conditions. The methodology is robust to noise and provides stable recovery in the presence of noise. The paper introduces a framework for proving the main result, which involves establishing approximate $ \ell_1 $ isometries and approximate dual certificates. The main result is that the convex program recovers the signal exactly (up to a global phase) provided the number of magnitude measurements is on the order of $ n \log n $. The paper also discusses the geometry of the problem and the stability of the method in the presence of noise. The paper concludes with a discussion of the organization of the paper and the proofs of the main results.PhaseLift is a method for exactly and stably recovering signals from magnitude measurements using convex programming. The paper proves that if vectors $ z_i $ are sampled independently and uniformly at random on the unit sphere, a signal $ x \in \mathbb{C}^n $ can be recovered exactly (up to a global phase factor) by solving a semidefinite program—a trace-norm minimization problem. This holds with high probability when the number of measurements $ m $ is on the order of $ n \log n $. The method is robust to additive noise, and the recovery error is small when the data are corrupted by noise. The paper also shows that the combinatorial phase retrieval problem can be solved by convex programming techniques under certain conditions. The methodology is robust to noise and provides stable recovery in the presence of noise. The paper introduces a framework for proving the main result, which involves establishing approximate $ \ell_1 $ isometries and approximate dual certificates. The main result is that the convex program recovers the signal exactly (up to a global phase) provided the number of magnitude measurements is on the order of $ n \log n $. The paper also discusses the geometry of the problem and the stability of the method in the presence of noise. The paper concludes with a discussion of the organization of the paper and the proofs of the main results.