  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} close to
the true density matrix at an accelerated 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.
