Repetition code applied to decoherence-free subspaces

Question

The repetition code (figure below) can be used to detect bit-flip errors. Such a code is useful for studying, but it can't be used in more general scenarios where also phase errors occur.

I was wondering whether it is possible to combine a simple code, like this one, together with a decoherence-free subspace (DFS).

For example the one adopted in here encodes a qubit within 2 ions, preventing information to decay because of phase errors.

Can I (straightforwardly) work with a 2-layer stratification of the logical qubit, or rather I would mess it all up?

Answer 1

As mentioned in the paper you linked, using a DFS does not make the logical qubit immune to all phase errors. It is an error suppression technique. You can then try to append this to a repetition code, but as your repetition code grows, you're also more susceptible to phase errors.