Implementation of a three-qubit Toffoli gate in a single step

Single-step implementations of multiqubit gates are generally believed to provide a simpler design, a faster operation, and a lower decoherence. For three coupled qubits interacting with a photon field, a realizable scheme for a single-step Toffoli gate is investigated. We find that the three-qubit system can be described by four effective modified Jaynes-Cummings models in the states of two control qubits. Within the rotating-wave approximation, the modified Jaynes-Cummings models are shown to be reduced to the conventional Jaynes-Cummings models with renormalized couplings between qubits and photon fields. A single-step Toffoli gate is shown to be realizable with tuning of the four characteristic oscillation periods that satisfy a commensurate condition. Possible values of the system parameters are estimated for a single-step Toffoli gate. Further, we discuss our single-step Toffoli-gate operation errors due to imperfections in the system parameters; numerical calculations show that a Toffoli gate with high fidelity can be obtained by adjusting pairs of the photon-qubit and qubit-qubit coupling strengths. In addition, a decoherence effect on the Toffoli-gate operation due to a thermal reservoir is discussed.