# Clifford group (quantum gates) The group of gates that stabilizes every n-qubits Pauli gate i.e., $CPC^\dagger\in Pauli$ . It is generated by: [[Hadamard Gate (H-gate)| Hadamard gate]], [[S phase gate]], [[CNOT gate]]. ## Clifford operators a unitary operator $U$ is a Clifford operator if for every [[Pauli operators(gates) and Pauli group]] $P$, $UPU^\dagger$ is also a [[Pauli operators(gates) and Pauli group]]. It satisfies: $UP=U^\dagger P^\prime$ . Example: $H$ is Clifford because $ \begin{align} HXH^\dagger&=HXH=Z\in Pauli \\ HZH&=X\in Pauli\\ HYH&=-Y\in Pauli \end{align} $ ## CNOT is Clifford We will see what CNOT does on the generators of Pauli i.e., $X\otimes I,I\otimes X,Z\otimes I, I\otimes Z$: $ \begin{align} CNOT(X\otimes I)CNOT^\dagger|a,b\rangle&=CNOT(X\otimes I)CNOT|a,b\rangle\\ &=CNOT(X\otimes I)|a,a\oplus b\rangle\\ &=CNOT|a\oplus1,a\oplus b\rangle\\ &=|a\oplus1,b\oplus 1\rangle\\ &=(X\otimes X)|a,b\rangle \end{align} $ and $(X\otimes X)\in Pauli$. ## Classically simulated Any quantum circuit compose of only Clifford gates can be efficiently simulated classically. This is known as Gottesman-Knill theorem. The idea is related to the fact that n-qubit state can be represented by the n independent Pauli operators which generate the stabilizer group of the state. ## Created 2026-02-22 12:34