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I was reading this blog post and as I understood, a unitary operator is logical if and only if it's in $\mathcal{N}(\mathcal{S})$, and a Pauli operator is Logical if and only if it's in $\mathcal{C}(\mathcal{S})$, and for stabilizer group $\mathcal{S}$, $\mathcal{N}(\mathcal{S})=\mathcal{C}(\mathcal{S})$, because it's abelian by definition. So, are logical gates and Pauli operators the same thing?

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Let $\mathcal{S}$ be the set of stabilizers. It is not true that $N(\mathcal{S}) = C(\mathcal{S})$.

There can exist an operator $U$ such that for all $S\in \mathcal{S}$, $USU^\dagger = S'$ for some $S'\in \mathcal{S}$ but $S'\neq S$. Such an operator $U$ is an element of $N(\mathcal{S})$ but not $C(\mathcal{S})$ and it is a logical operation.

However, such a $U$ cannot be a product of Pauli operators because Pauli operators must either commute or anticommute with stabilizers. If $U$ is a Pauli product, you either have $USU^\dagger = S$ which means $U\in C(\mathcal{S})$ or $USU^\dagger = -S$ in which case $U$ is not a logical operation.

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