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A transversal logical gate for an $ n $ qubit code is a gate from the group of local unitaries $$ \bigotimes_{i=1}^n U(2) $$ which also preserves the codespace. For an $ ((n,K,d)) $ code we say a particular logical gate $ \tilde{g} \in U(K) $ can be implemented transversally if there exists some gate $$ g=\bigotimes_{i=1}^n g_i \in \bigotimes_{i=1}^n U(2) $$ which implements $ \tilde{g} $ when restricted to the codespace. If all the $ g_i $ are equal then we say $ \tilde{g} $ is strongly transversal.

My question is the following: For every $ [[n,1,d]] $ code does there exist some basis for the codespace with respect to which logical $ X $ and logical $ Z $ have a strongly transversal implementation?

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The [[4,1,2]] surface code, or any code with an even number of data qubits, either doesn't have a transversal X or doesn't have a transversal Z. Because logical X has to anticommute with logical Z, but pairs of X commute with pairs of Z.

They still have constant-depth Pauli gates, they just aren't done by broadcasting the physical operation over all the data qubits.

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  • $\begingroup$ I think there is a misunderstanding here. In my definition of transversality I do not require that all $ g_i $ are equal. Although I gave my definition in the question I perhaps should have stated more emphatically that the $ g_i $ do not need to all be equal. Your definition is what I would call "strict" or "strong" transversality. For example see this paper arxiv.org/pdf/2210.14066.pdf Definition 3. And indeed the [[4,1,2]] code is the first example that comes to mind for me as well of a code with Paulis not strongly transversal. However this is not the question I am asking. $\endgroup$ Commented Mar 21, 2023 at 14:39
  • $\begingroup$ Take the [4,1,2] surface code, for example with stabilizer $ S=<XXXX,ZZII,IIZZ> $ then logical $ X $ can be implemented transversally as $ XXII $ and logical $ Z $ can be implemented transversally as $ ZIZI $. Indeed, as I note in my question, the Pauli group can be implemented transversally for every stabilizer code using Pauli gates from the normalizer $ N(S) $. I'll add these comments to my question to clarify. $\endgroup$ Commented Mar 21, 2023 at 14:44
  • $\begingroup$ Alternatively I suppose I could just change this question to be "Does every code have a strongly transversal Pauli group?", accept this answer and create a new question called "Does every code have a transversal Pauli group?" $\endgroup$ Commented Mar 21, 2023 at 15:00
  • $\begingroup$ Ok I just decided to do that, put in edits to the title and body of the question asking for strong transversality and make a new question here quantumcomputing.stackexchange.com/questions/31752/… Nice answer! $\endgroup$ Commented Mar 21, 2023 at 15:31

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