Clifford gates

The Pauli matrices,

${\displaystyle \sigma _{0}={\begin{pmatrix}1&0\\0&1\end{pmatrix}},\quad \sigma _{1}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad \sigma _{2}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},{\text{ and }}\sigma _{3}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}$

provide a basis for the density operators of a single qubit, as well as for the unitaries that can be applied to them. For the ${\displaystyle n}$-qubit case, one can construct a group, known as the Pauli group, according to

${\displaystyle \mathbf {P} _{n}=\left\{e^{i\theta \pi /2}\sigma _{j_{1}}\otimes \cdots \otimes \sigma _{j_{n}}\mid \theta =0,1,2,3,j_{k}=0,1,2,3\right\}.}$

The Clifford group is defined as the group of unitaries that normalize the Pauli group: ${\displaystyle \mathbf {C} _{n}=\{V\in U_{2^{n}}\mid V\mathbf {P} _{n}V^{\dagger }=\mathbf {P} _{n}\}.}$ The Clifford gates are then defined as elements in the Clifford group.

Some authors choose to define the Clifford group as the quotient group ${\displaystyle \mathbf {C} _{n}/U(1)}$. For ${\displaystyle n=}$ 1, 2, and 3, this group contains 24, 11,520, and 92,897,280 elements, respectively. [2]

Quantum circuits constructed from Clifford gates can be efficiently simulated with a classical computer, a result commonly known as the Gottesman–Knill theorem.

• Magic state distillation
• Clifford algebra