In physics and mathematics, the Pauli group
G
1
on 1 qubit is the 16-element matrix group consisting of the 2 × 2 identity matrix
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
)
,
together with the products of these matrices with the factors
−
1
and
±
i
:
G
1
=
d
e
f
{
±
I
,
±
i
I
,
±
X
,
±
i
X
,
±
Y
,
±
i
Y
,
±
Z
,
±
i
Z
}
≡
⟨
X
,
Y
,
Z
⟩
.
The Pauli group is generated by the Pauli matrices, and like them it is named after Wolfgang Pauli.
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 Hilbert space
(
C
2
)
⊗
n
.
As an abstract group,
G
1
≅
C
4
∘
D
4
is the central product of a cyclic group of order 4 and the dihedral group of order 8.
