This paper introduces a method for quantum state tomography based on compressed sensing, which significantly reduces the number of measurements required to reconstruct a quantum state. The method is particularly effective for quantum states that are nearly pure or have low rank, requiring only $ O(rd \log^{2}d) $ measurements, compared to the standard $ d^2 $ measurements needed for general states. The approach uses simple Pauli measurements, fast convex optimization, and is robust to noise. It allows for the reconstruction of a density matrix with a guaranteed low-rank approximation, and can be applied to states that are only approximately low-rank. The method also enables certified tomography of nearly pure states without prior assumptions about their properties. The paper provides theoretical bounds and numerical simulations demonstrating the effectiveness of the method. The approach is also applicable to process tomography, where it can characterize unknown quantum processes by performing state tomography on the corresponding Jamiołkowski state. The method is efficient and practical, with fast algorithms for reconstructing the density matrix from measurement data, and has been shown to work well in both theoretical and numerical simulations. The results demonstrate that the method can significantly reduce the experimental complexity and improve the efficiency of quantum state tomography for realistic scenarios.This paper introduces a method for quantum state tomography based on compressed sensing, which significantly reduces the number of measurements required to reconstruct a quantum state. The method is particularly effective for quantum states that are nearly pure or have low rank, requiring only $ O(rd \log^{2}d) $ measurements, compared to the standard $ d^2 $ measurements needed for general states. The approach uses simple Pauli measurements, fast convex optimization, and is robust to noise. It allows for the reconstruction of a density matrix with a guaranteed low-rank approximation, and can be applied to states that are only approximately low-rank. The method also enables certified tomography of nearly pure states without prior assumptions about their properties. The paper provides theoretical bounds and numerical simulations demonstrating the effectiveness of the method. The approach is also applicable to process tomography, where it can characterize unknown quantum processes by performing state tomography on the corresponding Jamiołkowski state. The method is efficient and practical, with fast algorithms for reconstructing the density matrix from measurement data, and has been shown to work well in both theoretical and numerical simulations. The results demonstrate that the method can significantly reduce the experimental complexity and improve the efficiency of quantum state tomography for realistic scenarios.