  Universal quantum computation on decoherence-free subspaces and subsystems
(DFSs) is examined with particular emphasis on using only physically relevant
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 Hamiltonians whose
induced evolution always preserves a DFS are derived within this stabilizer
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 computation which
always preserves the DFS (*natural fault-tolerant computation*) can be
performed using only two-body interactions. This is in marked contrast to
standard error correcting codes, where all known constructions using one or
two-body interactions must leave the codespace during the on-time of the
fault-tolerant gates. A further consequence of our universality construction 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 quantum systems,
is thus *asymptotically universal*.
