The paper presents a heralded protocol for generating distributed entanglement between error-protected logical qubits, which is crucial for fault-tolerant distributed quantum computing. The protocol uses a high-dimensional single photon to evolve physical qubits into logical qubits, entangling them with the photon. This entanglement is then converted into entanglement between two spatially separated logical qubits. The success of the entanglement generation is indicated by the detection of the photon, achieving near-unity efficiency. The logical qubit is encoded using a surface code consisting of four physical qubits, each implemented by a natural or artificial atom coupled to an optical cavity, providing robust protection against single-qubit errors. The protocol significantly reduces the number of quantum entangling gates needed and offers potential improvements in the scalability of quantum networks by reducing error accumulation and simplifying error correction. The interface between single photons and individual physical qubits is realized through a dipole-coupled optical cavity, specifically a silicon vacancy center (SiV$^-$) in diamond coupled to a single-sided nanocavity, which exhibits long coherence times and strong interaction with single photons at low temperatures.The paper presents a heralded protocol for generating distributed entanglement between error-protected logical qubits, which is crucial for fault-tolerant distributed quantum computing. The protocol uses a high-dimensional single photon to evolve physical qubits into logical qubits, entangling them with the photon. This entanglement is then converted into entanglement between two spatially separated logical qubits. The success of the entanglement generation is indicated by the detection of the photon, achieving near-unity efficiency. The logical qubit is encoded using a surface code consisting of four physical qubits, each implemented by a natural or artificial atom coupled to an optical cavity, providing robust protection against single-qubit errors. The protocol significantly reduces the number of quantum entangling gates needed and offers potential improvements in the scalability of quantum networks by reducing error accumulation and simplifying error correction. The interface between single photons and individual physical qubits is realized through a dipole-coupled optical cavity, specifically a silicon vacancy center (SiV$^-$) in diamond coupled to a single-sided nanocavity, which exhibits long coherence times and strong interaction with single photons at low temperatures.