physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

and mathematics, the Pauli group

G_1

on 1

qubit In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...

is the 16-element

matrix group In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a fa ...

consisting of the 2 × 2 identity matrix

I

and all of the

Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used ...

X = \sigma_1 = 
\begin
0&1\\
1&0
\end,\quad
Y = \sigma_2 = 
\begin
0&-i\\
i&0
\end,\quad
Z = \sigma_3 =
\begin
1&0\\
0&-1
\end

, together with the products of these matrices with the factors

\pm 1

and

\pm i

: :

G_1 \ \stackrel\   \ \equiv \langle X, Y, Z \rangle

. The Pauli group is generated by the Pauli matrices, and like them it is named after

Wolfgang Pauli Wolfgang Ernst Pauli (; ; 25 April 1900 – 15 December 1958) was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after having been nominated by Albert Einstein, Pauli received the Nobel Prize in Physics ...

. The Pauli group on

n

qubits,

G_n

, is the group generated by the operators described above applied to each of

n

qubits in the

tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...

Hilbert space

(\mathbb^2)^

. As an abstract group,

G_1\cong C_4 \circ D_4

is the

central product In mathematics, especially in the field of group theory, the central product is one way of producing a group from two smaller groups. The central product is similar to the direct product, but in the central product two isomorphic central subgroup ...

of a

cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...

of order 4 and the

dihedral group In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ...

of order 8.Pauli group o
GroupNames
/ref> The Pauli group is a representation of the

gamma group Gamma Group is an Anglo-German technology company that sells surveillance software to governments and police forces around the world. The company has been strongly criticised by human rights organisations for selling its FinFisher software to u ...

in three-dimensional Euclidean space. It is ''not'' isomorphic to the gamma group; it is less free, in that its chiral element is

\sigma_1\sigma_2\sigma_3=iI

whereas there is no such relationship for the gamma group.

References

External links

Finite groups Quantum information science 2. https://arxiv.org/abs/quant-ph/9807006 {{quantum-stub