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Recently there has been interest in the idea of quantum computing withoutcontrol of the physical interactions between component qubits. This is highlyappealing since the 'switching' of such interactions is a principal difficultyin creating real devices. It has been established that one can employ 'alwayson' interactions in a one-dimensional Heisenberg chain, provided that one cantune the Zeeman energies of the individual (pseudo-)spins. It is important togeneralize this scheme to higher dimensional networks, since a real devicewould probably be of that kind. Such generalisations have been proposed, butonly at the severe cost that the efficiency of qubit storage must *fall*. Herewe propose the use of multi-qubit gates within such higher-dimensional arrays,finding a novel three-qubit gate that can in fact increase the efficiencybeyond the linear model. Thus we are able to propose higher dimensionalnetworks that can constitute a better embodiment of the 'always on' concept - asubstantial step toward bringing this novel concept to full fruition.Comment: 20 pages in preprint format, inc. 3 figures. This version has fixed  typos and printer-friendly figures, and is to appear in NJ

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