We study efficient simulation of steady state for multi-dimensional rarefied gas flow, which is modeled by the Boltzmann equation with BGK-type collision term. A nonlinear multigrid solver is proposed to resolve the efficiency issue by the following approaches. The unified framework of numerical regularized moment method is first adopted to derive the high-quality discretization of the underlying problem. A fast sweeping iteration is introduced to solve the derived discrete problem more efficiently than the usual time-integration scheme on a single level grid. Taking it as the smoother, the nonlinear multigrid solver is then established to significantly improve the convergence rate. The OpenMP-based parallelization is applied in the implementation to further accelerate the computation. Numerical experiments for two lid-driven cavity flows and a bottom-heated cavity flow are carried out to investigate the performance of the resulting nonlinear multigrid solver. All results show the wonderful efficiency and robustness of the solver for both first- and second-order spatial discretization.