Quantum annealing with pairs of $^2\Sigma$ molecules as qubits

The rotational and fine structure of open-shell molecules in a $\Sigma$ electronic state gives rise to crossings between Zeeman states of different parity. These crossings become avoided in the presence of an electric field. We propose an algorithm that encodes Ising models into qubits defined by pairs of $^2\Sigma$ molecules sharing an excitation near these avoided crossings. This can be used to realize a transverse field Ising model tunable by an external electric or magnetic field, suitable for quantum annealing applications. We perform dynamical calculations for several examples with one- and two-dimensional connectivities. Our results demonstrate that the probability of obtaining valid annealing solutions is high and can be optimized by varying the annealing times.