Using the above potentials we demonstrate numerically the feasibility of the Rydberg annealer for the minimal instance (Fig. 1) with eight qubits and three ancillas. Figure 4 depicts the time dependent spectrum in reduced units for an instance of Hamiltonian equation (2) for random . The sweep functions At, Bt and Ct are simple linear functions. Note, that the efficiency can be considerably increased by adopting non-linear sweep functions. In Fig. 4 all energies are given relative to the ground state energy. The pronounced minimal gap is an order of magnitude smaller than the gap in the final state. Figure 4(b) shows the histogram of the success probability , defined as the overlap of the final state with the ground state , averaged over Nr=40 random instances for different sweep times far below the decoherence times τ<τ0/K. For the fastest switching time the average success probability is 75% and approaches unity for slower sweeps.