The most promising-seeming quantum error correction codes for the medium-to-long term are the topological codes, of which the toric code (and variants such as planar surface codes) and colour codes are the main examples.
As with essentially all approaches to quantum error correction, these are stabiliser codes. It also happens that both the toric code and colour codes are CSS codes: that is, they have a set of stabiliser generators which consist of products of $X$ operators or products of $Z$ operators, on subsets of the qubits. The useful properties of these codes for quantum error correction do not seem to particularly rely on this fact (the much more obviously relevant property is that the code distance of these codes scale with the size of the system and can be described by topological properties). So it may simply be a coincidence that these are CSS codes — but at the same time, it is intriguing that the codes with the best known thresholds seem to be CSS codes.
Question. Are there known examples of topological stabiliser codes which are neither CSS codes, nor equivalent to a CSS code up to local unitaries? Or is there a reason why such codes could not exist?
