Clifford gates

The Hadamard gate

${\displaystyle H={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}}$

is a member of the Clifford group as ${\displaystyle HXH^{\dagger }=Z}$ and ${\displaystyle HZH^{\dagger }=X}$.

The Phase gate

${\displaystyle S={\begin{bmatrix}1&0\\0&e^{i{\frac {\pi }{2}}}\end{bmatrix}}={\begin{bmatrix}1&0\\0&i\end{bmatrix}}={\sqrt {Z}}}$

is a Clifford gate as ${\displaystyle SXS^{\dagger }=Y}$ and ${\displaystyle SZS^{\dagger }=Z}$.

The CNOT gate applies to two qubits. It is a (C)ontrolled NOT gate, where a NOT gate is performed on qubit 2 if and only if qubit 1 is in the 1 state.

${\displaystyle CNOT={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{bmatrix}}.}$

Between ${\displaystyle X}$ and ${\displaystyle Z}$ there are four options:

CNOT Combinations

${\displaystyle P}$	CNOT ${\displaystyle P}$ CNOT${\displaystyle ^{\dagger }}$
${\displaystyle X\otimes I}$	${\displaystyle X\otimes X}$
${\displaystyle I\otimes X}$	${\displaystyle I\otimes X}$
${\displaystyle Z\otimes I}$	${\displaystyle Z\otimes I}$
${\displaystyle I\otimes Z}$	${\displaystyle Z\otimes Z}$

The CZ gate acts over two qubits. It is a (C)ontrolled Z gate, where effectively a Z gate is performed on qubit 2 if and only if qubit 1 is in the 1 state.

${\displaystyle CZ={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&-1\end{bmatrix}}.}$

The SWAP gate swaps two qubits,

${\displaystyle SWAP={\begin{bmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\end{bmatrix}}.}$