  We perform Monte Carlo simulations of the CP$^{N-1}$ model on the square
lattice for $N=10$, $21$, and $41$. Our focus is on the severe slowing down
related to instantons. To fight this problem we employ open boundary conditions
as proposed by L\"uscher and Schaefer for lattice QCD. Furthermore we test the
efficiency of parallel tempering of a line defect. Our results for open
boundary conditions are consistent with the expectation that topological
freezing is avoided, while autocorrelation times are still large. The results
obtained with parallel tempering are encouraging.
