Significance of Clifford operations from quantum error correction perspective

Question

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.

Possible answers:

1. 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.

2. 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.

Answer 1

Clifford operations are often easy to do fault-tolerantly in stabilizer codes, either transversally or by code deformation. The reason is exactly as you thought: the special relationship between these gates and the Paulis, since the latter are used to define stabilizer codes.

It is possible to get non-Clifford gates in codes, but a price must be paid. Specifically, there is a relationship between the geometric locality of codes and the gates they can do transversally. So if you are allowed to do only nearest neighbour controlled gates on a 2D lattice (such as a surface or Color code) only Cliffords will be possible. See papers like this one for more on this.

The fact that we can expect fault-tolerant Cliffords from stabilizer codes has subsequently been put at the heart of techniques to synthesize universal gate sets. So if there’s a way to create a non-stabilizer encoded state in a non-fault-tolerant way, we know how to clean it up using our logical Clifford’s. To turn these states into rotations, we use our logical Cliffords. So if you have a code and want to apply all these off-the-shelf results, you’d better find your fault-tolerant Cliffords. Or at least the Paulis, H and a CZ or CNOT if you can’t manage them all.