# In context of stabilizer codes, are logical gates and Pauli operators the same?

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?

## 1 Answer

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.