Quantum state tomography via non-convex Riemannian gradient descent

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.