  We study the conditions under which a subsystem code is correctable in the
presence of noise that results from continuous dynamics. We consider the case
of Markovian dynamics as well as the general case of Hamiltonian dynamics of
the system and the environment, and derive necessary and sufficient conditions
on the Lindbladian and system-environment Hamiltonian, respectively. For the
case when the encoded information is correctable during an entire time
interval, the conditions we obtain can be thought of as generalizations of the
previously derived conditions for decoherence-free subsystems to the case where
the subsystem is time dependent. As a special case, we consider conditions for
unitary correctability. In the case of Hamiltonian evolution, the conditions
for unitary correctability concern only the effect of the Hamiltonian on the
system, whereas the conditions for general correctability concern the entire
system-environment Hamiltonian. We also derive conditions on the Hamiltonian
which depend on the initial state of the environment, as well as conditions for
correctability at only a particular moment of time. We discuss possible
implications of our results for approximate quantum error correction.
