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I have read that logical gates can be implemented transversally ($U_{L} = V^{\otimes n}$) or weakly transversally ($U_{L} = \otimes _{i=1} ^{n} V_{i}$).

I have verified that the $[[8,3,2]]$ colour code has the logical gates: $$U_{1}=S_{1}S_{2}^{\dagger}S_{3}^{\dagger}S_{4}$$ $$U_{2}=S_{1}S_{2}^{\dagger}S_{5}^{\dagger}S_{6}$$ $$U_{3}=S_{1}S_{3}^{\dagger}S_{5}^{\dagger}S_{7}$$

$$A=T_{1}T_{2}^{\dagger}T_{3}^{\dagger}T_{4}T_{5}^{\dagger}T_{6}T_{7}T_{8}^{\dagger}.$$

I have read in several papers that the $3$ $U_{i}$ gates implement transversal CZ logical gates and the $A$ gate implements a weakly transversal CCZ logical gate.

However, from the definition of transversal $U_{L} = V^{\otimes n}$) and weakly transversal ($U_{L} = \otimes _{i=1} ^{n} V_{i}$), I don't understand why $U_{i}$ is transversal and $A$ is weakly transversal.

AG47
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am567
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1 Answers1

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It would be great if you could link the papers you are refering to.

I think that following these definitions, $U_1, U_2, U_3$ and $A$ are all weakly transversal but not transversal. You need two kinds of 1-qubit gates to implement $A$ and three kinds ($S, S^{\dagger}$ and $I$) to implement the others.

I would say that the confusion arised because a lot of (if not most) people call transversal an operation that is in fact only weakly transversal. The main interest for transversal gates comes from guaranteed fault tolerance, which is also true for weak transversality.

The stronger notion of transversality can be of use. The (strong) transversality of the logical $CNOT$ gate is a characterization of CSS stabilizer codes (see for example this answer) or it might be important when discussing hardware constraints. Note that the answer I linked uses "strictly transversal", implying that the sole "transversal" term carries a weaker meaning in some people's mind.

AG47
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