  Implementing a scalable quantum information processor using polar molecules
in optical lattices requires precise control over the long-range dipole-dipole
interaction between molecules in selected lattice sites. We present here a
scheme using trapped open-shell $^2\Sigma$ polar molecules that allows dipolar
exchange processes between nearest and next-nearest neighbors to be controlled
to construct a generalized transverse Ising spin Hamiltonian with tunable $XX$,
$YY$ and $XY$ couplings in the rotating frame of the driving lasers. The scheme
requires a moderately strong bias magnetic field with near-infrared light to
provide local tuning of the qubit energy gap, and mid-infrared pulses to
perform rotational state transfer via stimulated Raman adiabatic passage. No
interaction between qubits is present in the absence of the infrared driving.
We analyze the fidelity of the resulting two-qubit matchgate, and demonstrate
its robustness as a function of the driving parameters. We discuss a realistic
application of the system for universal matchgate quantum computing in optical
lattices.
