We propose a many-qubit network with cavity QED by encoding qubits in decoherence-free subspace, based on which we can implement many-logic-qubit conditional gates by means of cavity assisted interaction with single-photon pulses. Our scheme could not only resist collective dephasing errors, but also much reduce the implementational steps compared to conventional methods doing the same job, and we can also complete communications between two arbitrary nodes. We show the details by implementing a three-logic-qubit Toffoli gate, and explore the experimental feasibility and challenge based on currently achievable cavity QED technologies.
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