(color online) Average fidelities of the three-qubit quantum controlled-phase gate based on (a) random initial states method and (b) trace-operator method with one hundred of random initial states versus V_Rr/Ω and Ω/V_rr. For simplicity, we set Ω₁ = Ω₂ = Ω₃ = Ω₄ = Ω. (c) Fidelity of the four-qubit quantum controlled-phase gate with the initial state being set as . The dissipative rate γ/Ω = 10⁻³ is used.