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Consider the $ [[4,2,2]] $ stabilizer code with stabilizer generators $ XXXX,ZZZZ $. Logical $ X $ on the first qubit is $$ XIXI $$ Logical $ Z $ on the first qubit $$ ZZII $$ Logical $ X $ on the second qubit is $$ XXII $$ Logical $ Z $ on the second qubit is $$ ZIZI $$ It is also the case that $ SWAP_{2,3} $ implements logical SWAP (this isn't technically transversal, but it is a code automorphism).

For any single qubit Clifford gate $ C $ the physical gate $ C^{\otimes 4} $ preserves the code space, implementing some logical operation. For example $ H^{\otimes 4} $ and $ S^{\otimes 4} $ both implement logical gates. The specific gates are described on page 3 of https://arxiv.org/abs/1610.03507 with $ H^{\otimes 4} $ implementing logical $ H^{\otimes 2} $ times a logical $ SWAP $ and $ S^{\otimes 4} $ implementing logical $ CZ $ times a logical $ ZZ $.

Are there any other transversal gates besides these? Really my question is is there a good reference listing (or claiming to list) all transversal gates of the $ [[4,2,2]] $ code?

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  • $\begingroup$ As far as single qubit transversal gates go, example 5 of arxiv.org/pdf/0706.1382.pdf claims that the (local) automorphism group of the $[[4,2,2]]$ code is $\langle H^{\otimes 4}, P^{\otimes 4} \rangle$. $\endgroup$
    – Eric Kubischta
    Commented May 8, 2023 at 0:45

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A nice reference for transversal gates available in small codes is https://arxiv.org/abs/1912.10063. In Table 1, they mention that Paulis and CNOTs are available with the [[4,2,2]] code and provide some further references.

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    $\begingroup$ Thanks for the reference. The $ [[4,2,2]] $ code is stabilizer code and in particular a CSS code. Every stabilizer code has transversal Paulis and every CSS code has transversal CNOT. So this paper does not say anything interesting about transversality of the the $ [[4,2,2]] $ code beyond the fact that it is a CSS code $\endgroup$
    – Ian Gershon Teixeira
    Commented May 1, 2023 at 17:19
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    $\begingroup$ @IanGershonTeixeira if their search was in fact exhaustive then there are no other gates. I didn't read the full paper to see how good their method is. I would have given it more attention if they had more interesting results. For example for the [[8,3,3]] code they only found the Paulis... $\endgroup$
    – unknown
    Commented May 1, 2023 at 21:53
  • $\begingroup$ The method in the paper above uses machine learning and is likely a heuristic and not an exhaustive search. I think their approach is most useful when exhaustive search is not possible. For the $[[4,2,2]]$ you could exhaustively check all local Clifford gates with a computer; there are 'only' $24^4$ of them. Maybe disjointness could be used to show that all logical operators are Clifford? $\endgroup$
    – Jonas Anderson
    Commented May 3, 2023 at 4:03
  • $\begingroup$ For future reference, @JonasAnderson is using Theorem 13 from arxiv.org/pdf/quant-ph/9704043.pdf that states that the automorphism group of a $[[2m, 2m-2, 2]]$ code lies in the Clifford group. This will only be relevant for the single qubit transversal gates but indeed one can perform an exhaustive (finite) search in this case. $\endgroup$
    – Eric Kubischta
    Commented May 8, 2023 at 0:48

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