The paper presents an experimental demonstration of distributed quantum computing (DQC) across a network of two photonically interconnected trapped-ion modules, each containing a network qubit and a circuit qubit. The modules are separated by approximately 2 meters and use quantum gate teleportation (QGT) to enable all-to-all logical connectivity. The key achievement is the deterministic teleportation of a controlled-Z (CZ) gate between two circuit qubits in separate modules, achieving a fidelity of 86%. This is followed by the execution of Grover's search algorithm, which demonstrates the ability to perform distributed quantum circuits comprising multiple non-local two-qubit gates. The authors also implement distributed iSWAP and SWAP circuits, showing the capability to distribute arbitrary two-qubit operations. The use of photons as quantum information carriers makes this technique applicable to various platforms beyond trapped-ion quantum computers, paving the way for large-scale quantum computing. The paper discusses the technical details of the experimental setup, including the generation of remote entanglement, local mixed-species entangling gates, and conditional operations, and provides a comprehensive error budget for the CZ gate teleportation.The paper presents an experimental demonstration of distributed quantum computing (DQC) across a network of two photonically interconnected trapped-ion modules, each containing a network qubit and a circuit qubit. The modules are separated by approximately 2 meters and use quantum gate teleportation (QGT) to enable all-to-all logical connectivity. The key achievement is the deterministic teleportation of a controlled-Z (CZ) gate between two circuit qubits in separate modules, achieving a fidelity of 86%. This is followed by the execution of Grover's search algorithm, which demonstrates the ability to perform distributed quantum circuits comprising multiple non-local two-qubit gates. The authors also implement distributed iSWAP and SWAP circuits, showing the capability to distribute arbitrary two-qubit operations. The use of photons as quantum information carriers makes this technique applicable to various platforms beyond trapped-ion quantum computers, paving the way for large-scale quantum computing. The paper discusses the technical details of the experimental setup, including the generation of remote entanglement, local mixed-species entangling gates, and conditional operations, and provides a comprehensive error budget for the CZ gate teleportation.