Universal fault-tolerant quantum computation on decoherence-free subspaces

A general scheme to perform universal, fault-tolerant quantum computation w ithin decoherence-free subspaces (DFSs) is presented. At most two-qubit int eractions are required, and the system remains within the DFS throughout th e entire implementation of a quantum gate. We show explicitly how to perfor m universal computation on clusters of the four-qubit DFS encoding one logi cal qubit each under spatially symmetric (collective) decoherence. Our resu lts have immediate relevance to quantum computer implementations in which q uantum logic is implemented through exchange interactions, such as the rece ntly proposed spin-spin coupled quantum dot arrays and donor-atom arrays.