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 normalize the ''n''-qubit Pauli group, i.e., map tensor products of Pauli matrices to tensor products of Pauli matrices through conjugation. The notion was introduced by Daniel Gottesman and is named after the mathematician William Kingdon Clifford. Quantum circuits that consist of only Clifford gates can be efficiently simulated with a classical computer due to the Gottesman–Knill theorem.

Clifford group

Definition

The Pauli matrices,

: $\sigma_0=I=\begin 1 & 0 \\ 0 & 1 \end, \quad \sigma_1=X=\begin 0 & 1 \\ 1 & 0 \end, \quad \sigma_2=Y=\begin 0 & -i \\ i & 0 \end, \text \sigma_3=Z=\begin 1 & 0 \\ 0 & -1 \end$

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 $n$-qubit case, one can construct a group, known as the Pauli group, according to

: $\mathbf_n=\left\.$

The Clifford group is defined as the group of unitaries that normalize the Pauli group: $\mathbf_n=\.$ The Clifford gates are then defined as elements in the Clifford group.

Some authors choose to define the Clifford group as the quotient group $\mathbf_n/U(1)$, which counts elements in $\mathbf_n$ that differ only by an overall phase factor as the same element. For $n=$ 1, 2, and 3, this group contains 24, 11,520, and 92,897,280 elements, respectively.

It turns out that the quotient group $\mathbf_n/\mathbf_n$ is isomorphic to the $2n\times 2n$ symplectic matrices . In the case of a single qubit, each element in $\mathbf_1$ can be expressed as a matrix product $\mathbf\mathbf$, where $\mathbf\in\$ and $\mathbf\in\mathbf_1=\$. Here $H$ is the Hadamard gate, $S$ the phase gate, and $W=HS$ and $V=W^$, $HS$ swap the axes as $WXV = Y$, $WYV = Z$ and $WZV = X$. For the remaining gates, $HV=R_x(-\pi/2)$ is a rotation along the x-axis, and $HW=S \sim R_Z(\pi/2)$ is a rotation along the z-axis.

Generators

The Clifford group is generated by three gates, Hadamard, ''S'' and CNOT gates. 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 $Y$ gate is equal to the product of $X$ and $Z$ gates. To show that a unitary $U$ is a member of the Clifford group, it suffices to show that for all $P \in \mathbf_n$ that consist only of the tensor products of $X$ and $Z$, we have $UPU^\dagger \in \mathbf_n$.

Hadamard gate

The Hadamard gate

:$H = \frac \begin 1 & 1 \\ 1 & -1 \end$

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

''S'' gate

The phase gate

:$S = \begin 1 & 0 \\ 0 & e^ \end = \begin 1 & 0 \\ 0 & i \end = \sqrt$

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

CNOT gate

The CNOT gate applies to two qubits. Between $X$ and $Z$ there are four options:

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
:$A=(X \otimes Z)CZ$.
We know that:
$CZ(X \otimes I)CZ^\dagger =X \otimes Z$.
If we multiply by ''CZ'' from the right
:$CZ(X \otimes I) =(X \otimes Z)CZ$.
So ''A'' is equivalent to
:$A=(X \otimes Z)CZ = CZ(X \otimes I)$.

Simulatability

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

# Preparation of qubits in computational basis states,
# Clifford gates, and
# 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 shift gate (historically known as the $\pi /8$ gate):

: T = \begin 1 & 0 \\ 0 & e^ \end = \sqrt = \sqrt /math>.

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

:TX = \left \rightleft \rightleft \right= \left \rightnot \in

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

See also

* Magic state distillation
* Clifford algebra

References

{{quantum computing

Quantum information science