  Solid state quantum computing proposals rely on adiabatic operations of the
exchange gate among localized spins in nanostructures. We study corrections to
the Heisenberg interaction between lateral semiconductor quantum dots in an
external magnetic field. Using exact diagonalization we obtain the regime of
validity of the adiabatic approximation. We also find qualitative corrections
to the Heisenberg model at high magnetic fields and in looped arrays of spins.
Looped geometries of localized spins generate flux dependent, multi-spin terms
which go beyond the basic Heisenberg model.
