  We report on the implementation of arbitrary circuits on a universal
two-qubit register that can act as the computational module in a trapped-ion
quantum computer based on the quantum charge-coupled device architecture. A
universal set of quantum gates is implemented on a two-ion Coulomb crystal of
$^9$Be$^+$ ions using only chip-integrated microwave addressing. Individual-ion
addressing is implemented using microwave micromotion sideband transitions; we
obtain upper limits on addressing cross-talk in the register. Arbitrary
two-qubit operations are characterized using the cycle benchmarking protocol.
