  High-efficiency quantum information processing is equivalent to the fewest
quantum resources and the simplest operations by means of logic qubit gates.
Based on the reflection geometry of a single photon interacting with a
three-level $\Lambda$-typle atom-cavity system, we present some refined
protocols for realizing controlled-not (CNOT), Fredkin, and Toffoli gates on
hybrid systems. The first control qubit of our gates is encoded on a flying
photon, and the rest qubits are encoded on the atoms in optical cavity.
Moreover, these quantum gates can be extended to the optimal synthesis of
multi-qubit CNOT, Fredkin and Toffoli gates with O(n) optical elements without
auxiliary photons or atoms. Further, the simplest single-qubit operations are
applied to the photon only, which make these logic gates experimentally
feasible with current technology.
