Clifford gates

The Pauli matrices,

${\displaystyle \sigma _{0}=I={\begin{bmatrix}1&0\\0&1\end{bmatrix}},\quad \sigma _{1}=X={\begin{bmatrix}0&1\\1&0\end{bmatrix}},\quad \sigma _{2}=Y={\begin{bmatrix}0&-i\\i&0\end{bmatrix}},{\text{ and }}\sigma _{3}=Z={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}}$

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 ${\displaystyle \mathbf {C} _{n}/U(1)}$, which counts elements in ${\displaystyle \mathbf {C} _{n}}$ that differ only by an overall phase factor as the same element. For ${\displaystyle n=}$ 1, 2, and 3, this group contains 24, 11,520, and 92,897,280 elements, respectively. [2]

It turns out[3] that the quotient group ${\displaystyle \mathbf {C} _{n}/\mathbf {P} _{n}}$ is isomorphic to the ${\displaystyle 2n\times 2n}$ symplectic matrices Sp(2n). In the case of a single qubit, each element in ${\displaystyle \mathbf {C} _{1}}$ can be expressed as a matrix product ${\displaystyle \mathbf {A} \mathbf {B} }$, where ${\displaystyle \mathbf {A} \in \{I,V,W,H,HV,HW\}}$ and ${\displaystyle \mathbf {B} \in \mathbf {P} _{1}=\{I,X,Y,Z\}}$. Here ${\displaystyle H}$ is the Hadamard gate, ${\displaystyle S}$ the phase gate, and ${\displaystyle W=HS}$ and ${\displaystyle V=W^{\dagger }}$, ${\displaystyle HS}$ swap the axes as ${\displaystyle WXV=Y}$, ${\displaystyle WYV=Z}$ and ${\displaystyle WZV=X}$. For the remaining gates, ${\displaystyle HV=R_{x}(-\pi /2)}$ is a rotation along the x-axis, and ${\displaystyle HW=S\sim R_{Z}(\pi /2)}$ is a rotation along the z-axis.

The Clifford group is generated by three gates, Hadamard, S and CNOT gates.[4][5][6] Since all Pauli matrices can be constructed from the phase S and Hadamard gates, each Pauli gate is also trivially an element of the Clifford group.

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 ${\displaystyle UPU^{\dagger }\in \mathbf {P} _{n}}$.

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. 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 Gottesman–Knill theorem states that a quantum circuit using only the following elements can be simulated efficiently on a classical computer:

1. Preparation of qubits in computational basis states,
2. Clifford gates, and
3. Measurements in the computational basis.

The Gottesman–Knill theorem shows that even some highly entangled states can be simulated efficiently. Several important types of quantum algorithms use only Clifford gates, most importantly the standard algorithms for entanglement distillation and for quantum error correction.