Moving Horizon Estimation (MHE) is a general method employed in many dynamic systems to monitor unmeasurable states. MHE can handle unavoidable physical constraints on the system by a constrained nonlinear optimization problem. However, since this approach requires repeated solving of the optimization problem, it is usually limited to slow-evolving, quasi-linear, low-order systems. In this work, we propose a method that accelerates the optimization procedure. To achieve this goal, Carleman linearization technique is employed to obtain a linear representation of a generic nonlinear system. Then, the sensitivity of the estimation error, gradient vector and Hessian matrix of the objective function are analytically derived. This information about the objective function significantly reduces computational costs and errors associated with numerical approximations of derivatives. Even though the representation appears linear, it is in fact a higher order approximation. Simulation results for a crystallization process show the efficiency and performance of the designed observer.