A set of universal quantum gates is a vital part of the theory of quantum computing, but is absent in the developing theory of Relativistic Quantum Information (RQI). Yet, the Unruh--DeWitt (UDW) detector formalism can be elevated to unitary gates between qubits and quantum fields and has allowed RQI applications in quantum Shannon theory, such as mutual information, coherent information, and quantum capacity in field-mediated quantum channels. Recently, experimental realizations of UDW-style qubits have been proposed in two-dimensional quantum materials, but their value as a quantum technology, including quantum communication and computation, is not yet clear, especially since fields introduce many avenues for decoherence. We introduce controlled-unitary UDW logic gates between qubit and field that are comparable to the two-qubit CNOT gate. We then extend this formalism to demonstrate Quantum State Transfer (QST) (two CNOT gates) and SWAP (three CNOT gates) channels. We illustrate the performance of these quantum operation gates with the diamond distance, a measure of distinguishability between quantum channels. Distinguishability measures like diamond distance allow for a rigorous comparison between field-mediated transduction through UDW detectors and local quantum mechanical operations and so quantify the performance of UDW detectors in quantum technological applications. Using the controlled-unitary qubit-field interactions we define an exact form of the CNOT gate. With this technique we also define quantum field-mediated single qubit operations associated with the Hadamard H, the S, and T gates. Thus, UDW detectors in simple settings enable a collection of gates known to provide universal quantum computing.