What are the transversal gates of the $[[4,2,2]]$ code?

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?

• 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 $\langle H^{\otimes 4}, P^{\otimes 4} \rangle$. Commented May 8, 2023 at 0:45

• 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 Commented May 1, 2023 at 17:19
• 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? Commented May 3, 2023 at 4:03
• 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. Commented May 8, 2023 at 0:48