State estimation has played an essential role in monitoring the operational states of power systems. However, lacking real-time measurements in distribution networks hinders the applications of traditional state estimation methods that require redundant measurement data. This article proposes an accelerated matrix completion method for distribution system state estimation under unobservable conditions. First, the matrix completion-based state estimation is established as an unconstrained optimization problem, in which the power flow constraints and the differences between estimated and measured data are formulated as penalty terms of the objective function. Then, to diminish the magnitude difference influence of various measurement types on state estimation, a weight matrix is designed to evaluate the relative error between estimated and measured data, which is different from the absolute errors considered in the existing matrix completion methods. Finally, an accelerated proximal alternating linearized minimization algorithm is developed to solve the designed matrix completion-based state estimation problem. Compared with the commonly used solution method, it utilizes closed-form solutions and extrapolation strategies to iteratively solve the matrix completion problem. The proposed method is tested on the IEEE 33-node and 118-node distribution networks. Simulation results show that the proposed method succeeds in solving state estimation problems under insufficient measurements, where the traditional state estimation methods are infeasible. Compared with the performance of existing matrix completion-based state estimation methods, the proposed method enhances estimation accuracy while significantly improving computational efficiency.