(color online) Average fidelities of the three-qubit quantum controlled-phase gate versus atomic spontaneous rate γ based on (a) trace-operator and (b) random initial states methods with one hundred of random initial states. For simplicity, we set Ω₁ = Ω₂ = Ω₃ = Ω, V_Rr = wΩ, Ω = qV_rr, and w = q without loss of generality.