Multi-qubit gates in arrays coupled by 'always-on' interactions

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 generalizations 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 towards bringing this novel idea to full fruition.