  The possible state space dimension increases exponentially with respect to
the number of qubits. This feature makes the quantum state tomography expensive
and impractical for identifying the state of merely several qubits. The recent
developed approach, compressed sensing, gives us an alternative to estimate the
quantum state with fewer measurements. It is proved that the estimation then
can be converted to a convex optimization problem with quantum mechanics
constraints. In this paper we present an alternating augmented Lagrangian
method for quantum convex optimization problem aiming for recovering pure or
near pure quantum states corrupted by sparse noise given observables and the
expectation values of the measurements. The proposed algorithm is much faster,
robust to outlier noises (even very large for some entries) and can solve the
reconstruction problem distributively. The simulations verify the superiority
of the proposed algorithm and compare it to the conventional least square and
compressive quantum tomography using Dantzig method.
