Clifford gates

The Clifford group is generated by three gates:

${\displaystyle {1}/{\sqrt {2}}}$

${\displaystyle i}$

${\displaystyle \mathbf {C} _{n}}$

${\displaystyle \mathbf {C} _{1}^{n}}$

The ${\displaystyle Y}$ gate is equal to the product of ${\displaystyle X}$ and ${\displaystyle Z}$ gates. To show that a unitary ${\displaystyle U}$ is a member of the Clifford group, it suffices to show that for all ${\displaystyle P\in \mathbf {P} _{n}}$ that consist only of the tensor products of ${\displaystyle X}$ and ${\displaystyle Z}$, we have ${\displaystyle UPU^{\dagger }\in \mathbf {P} _{n}}$.

Common generating gates

Hadamard gate

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 ${\displaystyle HZH^{\dagger }=X}$.

S gate

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 ${\displaystyle SZS^{\dagger }=Z}$.

CNOT gate

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.

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

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

CNOT combinations

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

Building a universal set of quantum gates