  We present a detailed study of quantum simulations of coupled spin systems in
surface-electrode ion-trap arrays, and illustrate our findings with a proposed
implementation of the hexagonal Kitaev model [A. Kitaev, Annals of Physics
321,2 (2006)]. The effective (pseudo)spin interactions making up such quantum
simulators are found to be proportional to the dipole-dipole interaction
between the trapped ions, and are mediated by motion which can be driven by
state-dependent forces. The precise forms of the trapping potentials and the
interactions are derived in the presence of a surface electrode and a cover
electrode. These results are the starting point to derive an optimized
surface-electrode geometry for trapping ions in the desired honeycomb lattice
of Kitaev's model, where we design the dipole-dipole interactions in a way that
allows for coupling all three bond types of the model simultaneously, without
the need for time discretization. Finally we propose a simple wire structure
that can be incorporated in a microfabricated chip to generate localized
state-dependent forces which drive the couplings prescribed by this particular
model; such a wire structure should be adaptable to many other situations.
