Precise reconstruction of unknown quantum states from measurement data, a process commonly called quantum state tomography, is a crucial component in the development of quantum information processing technologies. Many different tomography methods have been proposed over the years. Maximum-likelihood estimation is a prominent example, being the most popular method for a long period of time. Recently, more advanced neural-network methods have started to emerge. Here, we go back to basics and present a method for continuous-variable state reconstruction that is both conceptually and practically simple, based on convex optimization. Convex optimization has been used for process tomography and qubit-state tomography, but seems to have been overlooked for continuous-variable quantum-state tomography. We demonstrate high-fidelity reconstruction of an underlying state from data corrupted by thermal noise and imperfect detection, for both homodyne and heterodyne measurements. A major advantage over other methods is that convex optimization algorithms are guaranteed to converge to the optimal solution.
We propose a non-convex optimization algorithm, based on the Burer-Monteiro (BM) factorization, for the quantum process tomography problem, in order to estimate a low-rank process matrix χ for near-unitary quantum gates. In this work, we compare our approach against state of the art convex optimization approaches based on gradient descent. We use a reduced set of initial states and measurement operators that require 2 · 8ncircuit settings, as well as
O(4n) measurements for an underdetermined setting. We find our algorithm converges faster and achieves higher fidelities than state of the art, both in terms of measurement settings, as well as in terms of noise tolerance, in the cases of depolarizing and Gaussian noise models.