Defined inductively as given below.

Consider \(n\) qubits and the gate set \(PU(2^n)\).

\(\mathcal{C}^{(1)} \subseteq PU(2^n)\) be the group of Pauli operators. We then define

\[ \mathcal{C}^{(n)} = \{ U \ | \ UPU^* \in \mathcal{C}^{(n-1)},\ \forall P \in \mathcal{C}^{(1)} \} . \]

See [1] for a good reference.

References