Quantum state tomography via non-convex Riemannian gradient descent

Abstract

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.