quantum gate directory

The Pauli group on one qubit is defined as $\mathcal{P}_1 := \{ i^n \sigma_j \mid n, j \in \{0, 1, 2, 3\} \}$. The $n$-qubit Pauli group is defined by all of the tensor products of elements of $\mathcal{P}_1$.

$$\mathcal{P}_n = \left\{ \bigotimes\limits_{i=0}^{n-1} P_i \mid P_i \in \mathcal{P}_1 \right\}$$