  Electric field noise is a major source of motional heating in trapped ion
quantum computation. While the influence of trap electrode geometries on
electric field noise has been studied in patch potential and surface adsorbate
models, only smooth surfaces are accounted for by current theory. The effects
of roughness, a ubiquitous feature of surface electrodes, are poorly
understood. We investigate its impact on electric field noise by deriving a
rough-surface Green's function and evaluating its effects on adsorbate-surface
binding energies. At cryogenic temperatures, heating rate contributions from
adsorbates are predicted to exhibit an exponential sensitivity to local surface
curvature, leading to either a large net enhancement or suppression over smooth
surfaces. For typical experimental parameters, orders-of-magnitude variations
in total heating rates can occur depending on the spatial distribution of
absorbates. Through careful engineering of electrode surface profiles, our
results suggests that heating rates can be tuned over orders of magnitudes.
