Tomographic reconstruction of multiqubit states by phase lift with the Nesterov algorithm

Quantum state tomography is of crucial importance in assessing the quality of quantum information processing devices. Tomography fundamentally requires both experiments and data processing resources, which increase exponentially with the size of qubits. We propose using phase lift to precisely characterize an unknown quantum state using only a small part of the data. We numerically reconstruct the density matrix of multiqubit pure states including the Greenberger-Horne-Zeilinger (GHZ) state, W-class state, cluster state, and random states. The Nesterov algorithm shows high accuracy and high speed for pure states especially for the specific entangled states with a large number of qubits. It shows the highest accuracy among the several specific algorithms, specially for larger number of qubits.