In physics and mathematics , the Pauli group
G
1
{\displaystyle G_{1}}
on 1 qubit is the 16-element matrix group consisting of the 2 × 2 identity matrix
I
{\displaystyle I}
and all of the Pauli matrices
X
=
σ
1
=
(
0
1
1
0
)
,
Y
=
σ
2
=
(
0
−
i
i
0
)
,
Z
=
σ
3
=
(
1
0
0
−
1
)
{\displaystyle X=\sigma _{1}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad Y=\sigma _{2}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\quad Z=\sigma _{3}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}
,
The Möbius–Kantor graph , the Cayley graph of the Pauli group
G
1
{\displaystyle G_{1}}
with generators X , Y , and Z
together with the products of these matrices with the factors
±
1
{\displaystyle \pm 1}
and
±
i
{\displaystyle \pm i}
:
G
1
=
d
e
f
{
±
I
,
±
i
I
,
±
X
,
±
i
X
,
±
Y
,
±
i
Y
,
±
Z
,
±
i
Z
}
≡
⟨
X
,
Y
,
Z
⟩
{\displaystyle G_{1}\ {\stackrel {\mathrm {def} }{=}}\ \{\pm I,\pm iI,\pm X,\pm iX,\pm Y,\pm iY,\pm Z,\pm iZ\}\equiv \langle X,Y,Z\rangle }
.
The Pauli group is generated by the Pauli matrices, and like them it is named after Wolfgang Pauli .
The Pauli group on
n
{\displaystyle n}
qubits,
G
n
{\displaystyle G_{n}}
, is the group generated by the operators described above applied to each of
n
{\displaystyle n}
qubits in the tensor product Hilbert space
(
C
2
)
⊗
n
{\displaystyle (\mathbb {C} ^{2})^{\otimes n}}
.
As an abstract group,
G
1
≅
C
4
∘
D
4
{\displaystyle G_{1}\cong C_{4}\circ D_{4}}
is the central product of a cyclic group of order 4 and the dihedral group of order 8.[1]
The Pauli group is a representation of the gamma group in three-dimensional Euclidean space. It is not isomorphic to the gamma group; it is less free, in that its chiral element is
σ
1
σ
2
σ
3
=
i
I
{\displaystyle \sigma _{1}\sigma _{2}\sigma _{3}=iI}
whereas there is no such relationship for the gamma group.