| Summary: | Recently, there has been interest in the idea of quantum computing <em>without</em> 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 <em>fall</em>. 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 <em>beyond</em> 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.
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