  Dipole-phonon quantum logic (DPQL) leverages the interaction between polar
molecular ions and the motional modes of a trapped-ion Coulomb crystal to
provide a potentially scalable route to quantum information science. Here, we
study a class of candidate molecular ions for DPQL, the cationic alkaline-earth
monoxides and monosulfides, which possess suitable structure for DPQL and can
be produced in existing atomic ion experiments with little additional
complexity. We present calculations of DPQL operations for one of these
molecules, CaO$^+$, and discuss progress towards experimental realization. We
also further develop the theory of DPQL to include state preparation and
measurement and entanglement of multiple molecular ions.
