Clifford gates

In quantum computing and quantum information theory, the Clifford gates are the elements of the Clifford group, a set of mathematical transformations which effect permutations of the Pauli operators. The notion was introduced by Daniel Gottesman and is named after the mathematician William Kingdon Clifford.^[1] Quantum circuits that consist only of Clifford gates can be efficiently simulated with a classical computer due to Gottesman–Knill theorem.

Definition of Clifford Group[]

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 ${\displaystyle \mathbf {C} _{n}/U(1)}$. For ${\displaystyle n=}$ 1, 2, and 3, this group contains 24, 11,520, and 92,897,280 elements, respectively. ^[2]

Generators of the Clifford group[]

The Clifford group is generated by three gates, Hadamard, CNOT, and the S gates.^[3] 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 ${\displaystyle UPU^{\dagger }\in \mathbf {P} _{n}}$.

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 ${\displaystyle SYS^{\dagger }=-X}$.

CNOT gate[]

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

CZ gate[]

The CZgate applies to two qubits. It can be constructed from CNOT and Hadamard gates.^[4] Between ${\displaystyle X}$ and ${\displaystyle Z}$ there are four options:

CZ Combinations

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

Properties and applications[]

The order of Clifford gates and Pauli gates can be interchanged. For example, this can be illustrated by considering the following operator on 2 qubits

${\displaystyle A=(X\otimes Z)CZ}$

We know that

${\displaystyle CZ(X\otimes I)CZ^{\dagger }=X\otimes Z}$

If we multiply by CZ from the right

${\displaystyle CZ(X\otimes I)=(X\otimes Z)CZ}$

So A is equivalent to

${\displaystyle A=(X\otimes Z)CZ=CZ(X\otimes I)}$

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.

Building a universal set of quantum gates[]

The Clifford gates do not form a universal set of quantum gates as not all gates are members of the Clifford group and some gates cannot be arbitrarily approximated with a finite set of operations. An example is the phase gate

${\displaystyle T={\begin{bmatrix}1&0\\0&e^{i{\frac {\pi }{4}}}\end{bmatrix}}={\sqrt {S}}={\sqrt[{4}]{Z}}}$.

To show that the ${\displaystyle T}$ gate does not map the Pauli-${\displaystyle X}$ gate to another Pauli matrix:

${\displaystyle TX{T^{\dagger }}=\left[{\begin{array}{*{20}{c}}1&0\\0&{e^{i{\frac {\pi }{4}}}}\end{array}}\right]\left[{\begin{array}{*{20}{c}}0&1\\1&0\end{array}}\right]\left[{\begin{array}{*{20}{c}}1&0\\0&{e^{-i{\frac {\pi }{4}}}}\end{array}}\right]=\left[{\begin{array}{*{20}{c}}0&{e^{-i{\frac {\pi }{4}}}}\\{e^{i{\frac {\pi }{4}}}}&0\end{array}}\right]\not \in {{\mathbf {P} }_{n}}}$

However, the Clifford group, when augmented with the ${\displaystyle T}$ gate, forms a universal quantum gate set for quantum computation.

See also[]

• Magic state distillation
• Clifford algebra

References[]