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High-efficiency quantum information processing is equivalent to the fewestquantum resources and the simplest operations by means of logic qubit gates.Based on the reflection geometry of a single photon interacting with athree-level $\Lambda$-typle atom-cavity system, we present some refinedprotocols for realizing controlled-not (CNOT), Fredkin, and Toffoli gates onhybrid systems. The first control qubit of our gates is encoded on a flyingphoton, and the rest qubits are encoded on the atoms in optical cavity.Moreover, these quantum gates can be extended to the optimal synthesis ofmulti-qubit CNOT, Fredkin and Toffoli gates with O(n) optical elements withoutauxiliary photons or atoms. Further, the simplest single-qubit operations areapplied to the photon only, which make these logic gates experimentallyfeasible with current technology.

Refined quantum gates for $\Lambda$-type atom-photon hybrid systems