  We numerically investigate the possibilities of driving quantum algorithms
with laser pulses in a register of ultracold NaCs polar molecules in a static
electric field. We focuse on the possibilities of performing scalable logical
operations by considering circuits that involve intermolecular gates
(implemented on adjacent interacting molecules) to enable the transfer of
information from one molecule to another during conditional laser-driven
population inversions. We study the implementation of an arithmetic operation
(the addition of 0 or 1 on a binary digit and a carry in) which requires
population inversions only and the Deutsch-Josza algorithm which requires a
control of the phases. Under typical experimental conditions, our simulations
show that high fidelity logical operations involving several qubits can be
performed in a time scale of a few hundred of microseconds, opening promising
perspectives for the manipulation of a large number of qubits in these systems.
