  We consider stabilizer measurements for surface codes with neutral atoms and
identify gate protocols that minimize logical error rates in the presence of a
fundamental error source -- spontaneous emission from Rydberg states. We
demonstrate that logical error rates are minimized by protocols that prevent
the propagation of Rydberg leakage errors and not by protocols that minimize
the physical two-qubit error rate. We provide laser-pulse-level gate protocols
to counter these errors. These protocols significantly reduce the logical error
rate for implementations of surface codes involving one or two species of
atoms. Our work demonstrates the importance of optimizing quantum gates for
logical errors in addition to gate fidelities and opens the way to the
efficient realization of surface codes with neutral atoms.
