User:Cetrau/sandbox

Definition
The Pauli matrices,


 * $$\sigma_0=I=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \quad \sigma_1=X=\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \quad \sigma_2=Y=\begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}, \text{ and } \sigma_3=Z=\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$$

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


 * $$\mathbf{P}_n=\left\{ e^{i\theta\pi/2} \sigma_{j_1} \otimes \cdots \otimes \sigma_{j_n} \mid \theta = 0,1,2,3,j_k = 0,1,2,3 \right\}.$$

The Clifford group is defined as the group of unitaries that normalize the Pauli group: $$\mathbf{C}_n=\{V\in U_{2^n}\mid V\mathbf{P}_nV^\dagger = \mathbf{P}_n\}.$$ This definition is equivalent to stating that the Clifford group consists of unitaries generated by the circuits using Hadamard, Phase, and CNOT gates.

Some authors choose to define the Clifford group as the quotient group $$\mathbf{C}_n/U(1)$$, which counts elements in $$\mathbf{C}_n$$ 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.

A third possible definition of the Clifford group is as the quotient group $$\mathbf{C}_n/\mathbf{P}_n$$. It turns out that the quotient group $$\mathbf{C}_n/\mathbf{P}_n$$ is isomorphic to the $$2n\times 2n$$ symplectic matrices $Sp(2n,2)$ over the field $$\mathbb{F}_2$$ of two elements. In the case of a single qubit, each element in $$\mathbf{C}_1$$ can be expressed as a matrix product $$\mathbf{A}\mathbf{B}$$, where $$\mathbf{A}\in\{I,V,W,H,HV,HW\}$$ and $$\mathbf{B}\in\mathbf{P}_1=\{I,X,Y,Z\}$$. Here $$H$$ is the Hadamard gate, $$S$$ the phase gate, and $$W=HS$$ and $$V=W^{\dagger}$$, $$ HS $$ 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.

Generating gate library
The Clifford group is generated by three gates, Hadamard, Phase gate S, and CNOT.

Circuit complexity
Arbitrary Clifford group element can be generated as a circuit with no more than $$O(n^2/ \log(n))$$ gates.

Properties
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:


 * 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.