Quantum computing in arrays coupled by “always-on” interactions

It has recently been shown that one can perform quantum computation in a Heisenberg chain in which the interactions are always on, provided that one can abruptly tune the Zeeman energies of the individual (pseudo)spins. Here we provide a more complete analysis of this scheme, including several generalizations. We generalize the interaction to an anisotropic form (incorporating the $XY$ interaction as a limit), providing a proof that a chain coupled in this fashion tends to an effective Ising chain in the limit of far off-resonant spins. We derive the primitive two-qubit gate that results from exploiting abrupt Zeeman tuning with such an interaction. We also demonstrate, via numerical simulation, that the same basic scheme functions in the case of smoothly shifted Zeeman energies. We conclude with some remarks regarding generalizations to two- and three-dimensional arrays.