  We explore the physics of the anisotropic compass model under the influence
of perturbing Heisenberg interactions and present the phase diagram with
multiple quantum phase transitions. The macroscopic ground state degeneracy of
the compass model is lifted in the thermodynamic limit already by infinitesimal
Heisenberg coupling, which selects different ground states with Z_2 symmetry
depending on the sign and size of the coupling constants --- then low energy
excitations are spin waves, while the compass states reflecting columnar order
are separated from them by a macroscopic gap. Nevertheless, nanoscale
structures relevant for quantum computation purposes may be tuned such that the
compass states are the lowest energy excitations, thereby avoiding decoherence,
if a size criterion derived by us is fulfilled.
