  We present an iterative method to solve the multipartite quantum state
estimation problem. We demonstrate convergence for any informationally complete
set of generalized quantum measurements in every finite dimension. Our method
exhibits fast convergence in high dimension and strong robustness under the
presence of realistic errors both in state preparation and measurement stages.
In particular, for mutually unbiased bases and tensor product of generalized
Pauli observables it converges in a single iteration. We show outperformance of
our algorithm with respect to the state-of-the-art of maximum likelihood
estimation methods both in runtime and fidelity of the reconstructed states.
