  With nowadays steadily growing quantum processors, it is required to develop
new quantum tomography tools that are tailored for high-dimensional systems. In
this work, we describe such a computational tool, based on recent ideas from
non-convex optimization. The algorithm excels in the compressed-sensing-like
setting, where only a few data points are measured from a low-rank or
highly-pure quantum state of a high-dimensional system. We show that the
algorithm can practically be used in quantum tomography problems that are
beyond the reach of convex solvers, and, moreover, is faster than other
state-of-the-art non-convex approaches. Crucially, we prove that, despite being
a non-convex program, under mild conditions, the algorithm is guaranteed to
converge to the global minimum of the problem; thus, it constitutes a provable
quantum state tomography protocol.
