  We propose a new quantum state reconstruction method that combines ideas from
compressed sensing, non-convex optimization, and acceleration methods. The
algorithm, called Momentum-Inspired Factored Gradient Descent (\texttt{MiFGD}),
extends the applicability of quantum tomography for larger systems. Despite
being a non-convex method, \texttt{MiFGD} converges \emph{provably} to the true
density matrix at a linear rate, in the absence of experimental and statistical
noise, and under common assumptions. With this manuscript, we present the
method, prove its convergence property and provide Frobenius norm bound
guarantees with respect to the true density matrix. From a practical point of
view, we benchmark the algorithm performance with respect to other existing
methods, in both synthetic and real experiments performed on an IBM's quantum
processing unit. We find that the proposed algorithm performs orders of
magnitude faster than state of the art approaches, with the same or better
accuracy. In both synthetic and real experiments, we observed accurate and
robust reconstruction, despite experimental and statistical noise in the
tomographic data. Finally, we provide a ready-to-use code for state tomography
of multi-qubit systems.
