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Quantum state tomography aims to estimate the state of a quantum mechanical system which is described by a trace one, Hermitian positive semidefinite complex matrix, given a set of measurements of the state. Existing works focus on estimating the density matrix that represents the state, using a compressive sensing approach, with only fewer measurements than that required for a tomographically complete set, with the assumption that the true state has a low rank. One very popular method to estimate the state is the use of the Singular Value Thresholding (SVT) algorithm. In this work, we present a machine learning approach to estimate the quantum state of n-qubit systems by unrolling the iterations of SVT which we call Learned Quantum State Tomography (LQST). As merely unrolling SVT may not ensure that the output of the network meets the constraints required for a quantum state, we design and train a custom neural network whose architecture is inspired from the iterations of SVT with additional layers to meet the required constraints. We show that our proposed LQST with very few layers reconstructs the density matrix with much better fidelity than the SVT algorithm which takes many hundreds of iterations to converge. We also demonstrate the reconstruction of the quantum Bell state from an informationally incomplete set of noisy measurements.