Scalable ion trap quantum computation in decoherence-free subspaces with pairwise interactions only

We show that universal ion trap computation can be performed on decoherence-free subspaces (DFS's) using only two-qubit operations. One logical qubit is minimally encoded into three physical qubits. The encoded qubit is in a DFS under collective decoherence. Encoded single- and two-qubit logical operations are implemented via the Sorensen-Molmer interaction. We show that alternation of the effective Hamiltonians for two particular phase configurations of control fields approximates an anisotropic exchange interaction that has recently been shown to be universal on these encodings. Physically realistic rapid alternation of the control fields also approximates evolution in a DFS. Our scheme is scalable in a recently proposed array-based architecture.