We propose an efficient quantum algorithm for simulating the dynamics of Hamiltonians the possess symmetries, where the evolution takes place in a specific sector of interest of the symmetry operator. We show that the proposed method is optimal, i.e., the gate and qubit costs scale with the norm of the Hamiltonian restricted to the sector of interest, a very attractive property to have especially in the NISQ era, where robust error correction schemes are out of the question.