This paper presents the first experimental demonstration of distributed quantum computing (DQC) across two trapped-ion modules separated by 2 meters. The modules contain network and circuit qubits, with the network qubits used to generate entanglement via a photonic link. This entanglement is then used to mediate quantum gate teleportation (QGT), enabling the deterministic teleportation of a controlled-Z (CZ) gate between circuit qubits in separate modules. The teleportation process achieves 86% fidelity, and the implementation of Grover's search algorithm demonstrates the first distributed quantum algorithm with multiple non-local two-qubit gates, achieving a 71% success rate. The study also demonstrates the implementation of distributed iSWAP and SWAP circuits, which require 2 and 3 instances of QGT, respectively. These results show the ability to distribute arbitrary two-qubit operations across a network of quantum processing modules. The work highlights the potential of photonic interconnects for scalable quantum computing, as they enable all-to-all connectivity and reconfigurability. The results demonstrate the feasibility of deterministic quantum gate teleportation across a quantum network, which is essential for scalable quantum computing. The study also identifies the main sources of error in the teleportation process, including local operations and remote entanglement generation. The results suggest that the technical limitations in the current implementation can be overcome, paving the way for large-scale quantum computing. The study also discusses the potential applications of DQC beyond trapped-ion quantum computers, including quantum repeaters and hybrid quantum computing platforms. The work provides a foundation for future research in distributed quantum computing and quantum networking.This paper presents the first experimental demonstration of distributed quantum computing (DQC) across two trapped-ion modules separated by 2 meters. The modules contain network and circuit qubits, with the network qubits used to generate entanglement via a photonic link. This entanglement is then used to mediate quantum gate teleportation (QGT), enabling the deterministic teleportation of a controlled-Z (CZ) gate between circuit qubits in separate modules. The teleportation process achieves 86% fidelity, and the implementation of Grover's search algorithm demonstrates the first distributed quantum algorithm with multiple non-local two-qubit gates, achieving a 71% success rate. The study also demonstrates the implementation of distributed iSWAP and SWAP circuits, which require 2 and 3 instances of QGT, respectively. These results show the ability to distribute arbitrary two-qubit operations across a network of quantum processing modules. The work highlights the potential of photonic interconnects for scalable quantum computing, as they enable all-to-all connectivity and reconfigurability. The results demonstrate the feasibility of deterministic quantum gate teleportation across a quantum network, which is essential for scalable quantum computing. The study also identifies the main sources of error in the teleportation process, including local operations and remote entanglement generation. The results suggest that the technical limitations in the current implementation can be overcome, paving the way for large-scale quantum computing. The study also discusses the potential applications of DQC beyond trapped-ion quantum computers, including quantum repeaters and hybrid quantum computing platforms. The work provides a foundation for future research in distributed quantum computing and quantum networking.