The possible state space dimension increases exponentially with respect tothe number of qubits. This feature makes the quantum state tomography expensiveand impractical for identifying the state of merely several qubits. The recentdeveloped approach, compressed sensing, gives us an alternative to estimate thequantum state with fewer measurements. It is proved that the estimation thencan be converted to a convex optimization problem with quantum mechanicsconstraints. In this paper we present an alternating augmented Lagrangianmethod for quantum convex optimization problem aiming for recovering pure ornear pure quantum states corrupted by sparse noise given observables and theexpectation values of the measurements. The proposed algorithm is much faster,robust to outlier noises (even very large for some entries) and can solve thereconstruction problem distributively. The simulations verify the superiorityof the proposed algorithm and compare it to the conventional least square andcompressive quantum tomography using Dantzig method.
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