It is well known that CSS codes can have lots of transversal entangling gates. For example, $ CNOT $ is exactly transversal on 2 blocks of any $ [[n,1,d]] $ CSS code. And https://arxiv.org/abs/1304.3709 shows that $ CCZ $ is exactly transversal on 3 blocks of any triorthogonal CSS code. Also $$ T \otimes T^\dagger \otimes T^\dagger \otimes T \otimes T^\dagger \otimes T \otimes T \otimes T^\dagger $$ implements logical $ CCZ $ on the $ [[8,3,2]] $ colour code, see https://earltcampbell.com/2016/09/26/the-smallest-interesting-colour-code/, and $ S^{\otimes 4} $ on the $ [[4,2,2]] $ code implements logical $ CZ $ followed by logical $ ZZ $, see https://arxiv.org/abs/1610.03507.

However I have never seen an example of a transversal gate implementing a logical entangling gate on a non-CSS stabilizer code.

Are transversal entangling gates possible for stabilizer codes other than CSS?



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