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For a $[[n, 1]]$ QEC code $\mathcal{Q}$, we say single logical gate $R$ is transversal if the logical $\bar{R}$ can be implemented with $R^{\otimes n}$.

I am wondering if we could expand the definition to $k$-local transversal: The logical operation can be implemented with a tensor product of $k$-subsystem operations. Therefore, we quickly have two special cases:

  1. $1$-local transversal: This coincides with the canonical definition of transversal (Note that I am only discussing single-qubit logical gates here).

  2. $n$-local transversal: For $n$-qubit system, $n$-local is global. This holds for any $n$-qubit stabilizer codes and for any single-qubit gates $R$ apparently.

Are there already any rigorous definition on this concept? If so, are there any examples for codes with useful $k$-local transversal operations?

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Not exactly the same as what you're asking, but related, is the concept used here: https://arxiv.org/abs/1206.1609 Once you go beyond transversal application of single-qubit operations, what becomes interesting is the depth of sequences of operations and how they interact with each other, propagating out operations/correlations in a sort of light cone.

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