Error-protected photonic logical cluster states generation based on photonic Faraday rotation in an atom-cavity coupling system

Logical entangled states are essential for quantum information processing due to their robustness against environmental noise and decoherence. We propose a scheme for generating logical cluster states based on photonic Faraday rotation in atom-cavity coupling systems. A physical cluster state is generated first. Subsequently, additional single photons assist in transforming each physical qubit into a physical GHZ state, which serves as a logical qubit. After replacing all physical qubits with logical qubits, the logical cluster state is obtained. The responses of photon detectors reveal low-probability errors arising from photon input-output imbalances, but these measurements do not induce scheme failure. Optical attenuators balance optical paths, guaranteeing unity fidelity in theory.