  The recovery of an unknown density matrix of large size requires huge
computational resources. The recent Factored Gradient Descent (FGD) algorithm
and its variants achieved state-of-the-art performance since they could
mitigate the dimensionality barrier by utilizing some of the underlying
structures of the density matrix. Despite their theoretical guarantee of a
linear convergence rate, the convergence in practical scenarios is still slow
because the contracting factor of the FGD algorithms depends on the condition
number $\kappa$ of the ground truth state. Consequently, the total number of
iterations can be as large as $O(\sqrt{\kappa}\ln(\frac{1}{\varepsilon}))$ to
achieve the estimation error $\varepsilon$. In this work, we derive a quantum
state tomography scheme that improves the dependence on $\kappa$ to the
logarithmic scale; namely, our algorithm could achieve the approximation error
$\varepsilon$ in $O(\ln(\frac{1}{\kappa\varepsilon}))$ steps. The improvement
comes from the application of the non-convex Riemannian gradient descent (RGD).
The contracting factor in our approach is thus a universal constant that is
independent of the given state. Our theoretical results of extremely fast
convergence and nearly optimal error bounds are corroborated by numerical
results.
