  Quantum error-correcting codes (QECCs) and decoherence-free subspace (DFS)
codes provide active and passive means, respectively, to address certain types
of errors that arise during quantum computation. The latter technique is
suitable to correct correlated errors with certain symmetries and the former to
correct independent errors. The concatenation of a QECC and a DFS code results
in a degenerate code that splits into actively and passively correcting parts,
with the degeneracy impacting either part, leading to degenerate errors as well
as degenerate stabilizer operators. The concatenation of the two types of code
can aid universal fault-tolerant quantum computation when a mix of correlated
and independent errors is encountered. In particular, we show that for
sufficiently strongly correlated errors, the concatenation with the DFS as the
inner code provides better entanglement fidelity, whereas for sufficiently
independent errors, the concatenation with the QECC as the inner code is
preferable. As illustrative examples, we examine in detail the concatenation of
a two-qubit DFS code and a three-qubit repetition code or five-qubit
Knill-Laflamme code, under independent and correlated errors.
