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What are examples of interesting $ [[n,1,d]] $ or $ [[n,2,d]] $ stabilizer codes, $ d \geq 2 $, whose group of transversal gates is not isomorphic to a subgroup of the Clifford group (on 1 and 2 qubits respectively).

The first example that comes to mind for me is the $ [[15,1,3]] $ code which has an element, the $ T $ gate, whose image in $ PU_2 $ has order 8 and thus cannot be embedded in the Clifford group.

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That is more or less is it for [[n,1,d]] codes: we know that the only non-Clifford gates you can have are, up to Clifford equivalences, phase gates of the form $2\pi/2^k$. So, if you take the construction for the [[15,1,3]] code and generalise it to $[[2^{k+1}-1,1,3]]$, you get T, square root of T, and so on. They also come with controlled-phase and controlled-controlled-phase etc (basically, for each control you add, the rotation angle doubles). There are some details (and more useful references) in a very recent paper of mine: arXiv:2210.14066.

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  • $\begingroup$ "we know that the only non-Clifford gates you can have are, up to Clifford equivalences, phase gates of the form $ 2 \pi/ 2^k $." That is very interesting to me, could you say more about that? What method can be used to show that transversal gates of stabilizer states must be of that form? $\endgroup$ Oct 31, 2022 at 22:38
  • $\begingroup$ ah I guess that you are referring to reference [21] of your paper: [21] J. T. Anderson and T. Jochym-O’Connor, “Classification of transversal gates in qubit stabilizer codes,” Not especially related, but something I noticed is that at the end of the first paragraph this reference mentions trying to implement the $ V $ gate transversally, but that is impossible by Eastin-Knill theorem since the $ V $ gate does not have finite order. $\endgroup$ Nov 1, 2022 at 0:45
  • $\begingroup$ It's not about implementing it as such. At least the way that I think about it is equivalence classes of codes. If I can transform one code into another code via the action of transversal Cliffords (most interestingly, when the operation is not a logical gate on the code), then they're equivalent, particularly with respect to their logical gate sets. $\endgroup$
    – DaftWullie
    Nov 1, 2022 at 8:52
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Might be worthwhile to mention that the stabilizer codes mentioned by DaftWullie go under the name of Reed-Muller codes and can be implemented with color codes in $D=2^{k+1}-1$ spatial dimensions (see for instance here). More generally, we know that the transversal gate sets of stabilizer codes (and those implementable by shallow circuits) lie in some finite level of the Clifford hierarchy (see here and references therein). For geometrically local codes in $D$ spatial dimensions, those sets are limited to the $D$-th level (ref) .

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