Theory of decoherence-free fault-tolerant universal quantum computation - art. no. 042307

Universal quantum computation on decoherence-free subspaces and subsystems (DFSs) is examined with particular emphasis on using only physically releva nt interactions. A necessary and sufficient condition for the existence of decoherence-free (noiseless) subsystems in the Markovian regime is derived here for the first time. A stabilizer formalism for DFSs is then developed which allows for the explicit understanding of these in their dual role as quantum error correcting codes. Conditions for the existence of Hamiltonian s whose induced evolution always preserves a DFS are derived within this st abilizer formalism. Two possible collective decoherence mechanisms arising from permutation symmetries of the system-bath coupling are examined within this framework. It is shown that in both cases universal quantum computati on which always preserves the DFS (natural fault-tolerant computation) can be performed using only two-body interactions. This is in marked contrast t o standard error correcting codes, where all known constructions using one- or two-body interactions must leave the code space during the on-time of t he fault-tolerant gates. A further consequence of our universality construc tion is that a single exchange Hamiltonian can be used to perform universal quantum computation on an encoded space whose asymptotic coding efficiency is unity. The exchange Hamiltonian, which is naturally present in many qua ntum systems, is thus asymptotically universal.