In the literature on QECC, Clifford gates occupy an elevated status.
Consider the following examples which attest to this:
When you study stabilizer codes, you separately study how to perform encoded Clifford gates (even if these aren't applicable transversally). All introductory material on QECC emphasize on performing encoded Clifford operations on quantum codes. And otherwise too, emphasize on Clifford gates (i.e., even when not performing encoded Clifford gates in quantum codes).
The entire topic of magic state distillation* is based on the classification of certain operations (including the performance of Clifford gates) as low-cost operations, while, for instance, performing the toffoli-gate or the $\pi/8$-gate, as higher-cost operations.
- This has been justified in certain places in the literature ,for e.g., Gottesman's PhD dissertation and many papers by him, and also in https://arxiv.org/abs/quant-ph/0403025. The reason given in these places is that it is possible to perform some Clifford gates transversally (a prototypical Fault-tolerant operation) on certain stabilizer codes. On the other hand, it is not easy to find a transversal application of non-Clifford gates on quantum codes. I haven't verified this myself, but am just going by statements which Gottesman makes in his PhD. dissertation and some review articles.
Not being able to perform an encoded gate transversally on a quantum code immediately increases the cost of performing said gate on the code. And hence performing Clifford gates goes into the low-cost category, while non-Clifford gates goes into the high-cost category.
- From an engineering perspective, it is important to decide on a standardized list of basic units of quantum computation (state preparation, gates, measurement-observables/basis), etc. Performing Clifford gates makes for a convenient choice on that list because of multiple reasons (most well-known sets of universal quantum gates include many Clifford gates in them ,Gottesman-Knill theorem**, etc).
These are the only two reasons I could think of for why the Clifford group has such an elevated status in the study of QECC (particularly when you're studying stabilizer codes). Both reasons stem from an engineering perspective.
So the question is can one identify other reasons, which don't stem from an engineering perspective? Is there some other major role that the Clifford gates play, which I've missed out?
Possible other reason: I know that the Clifford group is the normaliser of the Pauli group in the Unitary group (on $n$ qubit systems). Also, that it has a semidirect product structure (actually a projective representation of of semidirect product group). Does these relations/properties by themselves give another reason why one ought to study the Clifford group in association with Stabilizer codes?
*Feel free to correct this. **Which states that restricted to certain operations, you can't obtain the quantum advantage, and hence you need a little-bit more than the set of operations you initially restricted yourself to.