  We show that an array of polar molecules interacting with Rydberg atoms is a
promising hybrid system for scalable quantum computation. Quantum information
is stored in long-lived hyperfine or rotational states of molecules which
interact indirectly through resonant dipole-dipole interactions with Rydberg
atoms. A two-qubit gate based on this interaction has a duration of 1 $\mu$s
and an achievable fidelity of 99.9%. The gate has little sensitivity to the
motional states of the particles -- the molecules can be in thermal states, the
atoms do not need to be trapped during Rydberg excitation, the gate does not
heat the molecules, and heating of the atoms has a negligible effect. Within a
large, static array, the gate can be applied to arbitrary pairs of molecules
separated by tens of micrometres, making the scheme highly scalable. The
molecule-atom interaction can also be used for rapid qubit initialization and
efficient, non-destructive qubit readout, without driving any molecular
transitions. Single qubit gates are driven using microwave pulses alone,
exploiting the strong electric dipole transitions between rotational states.
Thus, all operations required for large scale quantum computation can be done
without moving the molecules or exciting them out of their ground electronic
states.
