The paper introduces a novel method for quantum state tomography based on compressed sensing, specifically tailored for pure or nearly pure quantum states. The method significantly reduces the number of measurement settings required compared to standard methods, which typically need \(d^2\) settings for a \(d\)-dimensional Hilbert space. The proposed approach uses only simple Pauli measurements and fast convex optimization, making it experimentally feasible. The method can handle states that are only approximately low-rank and is robust to noise. The authors provide theoretical bounds and numerical simulations to support their findings. They also present a hybrid approach that further optimizes the classical post-processing step, achieving faster and more accurate results. The method is applicable to both pure and mixed quantum states and can be used to characterize quantum processes.The paper introduces a novel method for quantum state tomography based on compressed sensing, specifically tailored for pure or nearly pure quantum states. The method significantly reduces the number of measurement settings required compared to standard methods, which typically need \(d^2\) settings for a \(d\)-dimensional Hilbert space. The proposed approach uses only simple Pauli measurements and fast convex optimization, making it experimentally feasible. The method can handle states that are only approximately low-rank and is robust to noise. The authors provide theoretical bounds and numerical simulations to support their findings. They also present a hybrid approach that further optimizes the classical post-processing step, achieving faster and more accurate results. The method is applicable to both pure and mixed quantum states and can be used to characterize quantum processes.