  We study the effect of continuous quantum error correction in the case where
each qubit in a codeword is subject to a general Hamiltonian interaction with
an independent bath. We first consider the scheme in the case of a trivial
single-qubit code, which provides useful insights into the workings of
continuous error correction and the difference between Markovian and
non-Markovian decoherence. We then study the model of a bit-flip code with each
qubit coupled to an independent bath qubit and subject to continuous
correction, and find its solution. We show that for sufficiently large
error-correction rates, the encoded state approximately follows an evolution of
the type of a single decohering qubit, but with an effectively decreased
coupling constant. The factor by which the coupling constant is decreased
scales quadratically with the error-correction rate. This is compared to the
case of Markovian noise, where the decoherence rate is effectively decreased by
a factor which scales only linearly with the rate of error correction. The
quadratic enhancement depends on the existence of a Zeno regime in the
Hamiltonian evolution which is absent in purely Markovian dynamics. We analyze
the range of validity of this result and identify two relevant time scales.
Finally, we extend the result to more general codes and argue that the
performance of continuous error correction will exhibit the same qualitative
characteristics.
