  Distributed quantum computing (DQC) combines the computing power of multiple
networked quantum processing modules, enabling the execution of large quantum
circuits without compromising on performance and connectivity. Photonic
networks are well-suited as a versatile and reconfigurable interconnect layer
for DQC; remote entanglement shared between matter qubits across the network
enables all-to-all logical connectivity via quantum gate teleportation (QGT).
For a scalable DQC architecture, the QGT implementation must be deterministic
and repeatable; until now, there has been no demonstration satisfying these
requirements. We experimentally demonstrate the distribution of quantum
computations between two photonically interconnected trapped-ion modules. The
modules are separated by $\sim$ 2 m, and each contains dedicated network and
circuit qubits. By using heralded remote entanglement between the network
qubits, we deterministically teleport a controlled-Z gate between two circuit
qubits in separate modules, achieving 86% fidelity. We then execute Grover's
search algorithm - the first implementation of a distributed quantum algorithm
comprising multiple non-local two-qubit gates - and measure a 71% success rate.
Furthermore, we implement distributed iSWAP and SWAP circuits, compiled with 2
and 3 instances of QGT, respectively, demonstrating the ability to distribute
arbitrary two-qubit operations. As photons can be interfaced with a variety of
systems, this technique has applications extending beyond trapped-ion quantum
computers, providing a viable pathway towards large-scale quantum computing for
a range of physical platforms.
