  Recently there has been interest in the idea of quantum computing without
control of the physical interactions between component qubits. This is highly
appealing since the 'switching' of such interactions is a principal difficulty
in creating real devices. It has been established that one can employ 'always
on' interactions in a one-dimensional Heisenberg chain, provided that one can
tune the Zeeman energies of the individual (pseudo-)spins. It is important to
generalize this scheme to higher dimensional networks, since a real device
would probably be of that kind. Such generalisations have been proposed, but
only at the severe cost that the efficiency of qubit storage must *fall*. Here
we propose the use of multi-qubit gates within such higher-dimensional arrays,
finding a novel three-qubit gate that can in fact increase the efficiency
beyond the linear model. Thus we are able to propose higher dimensional
networks that can constitute a better embodiment of the 'always on' concept - a
substantial step toward bringing this novel concept to full fruition.
