For qubit Pauli group is the finite
matrix group of Kronecker
products of the Pauli matrices . More explicitly,
on writing
Here, identity matrix
and imaginary
unit . This convention includes all four scalar phases. Hashagen et al. (2018)
call the subgroup generated by Kronecker
products of et al. (2021) call orders ,
so the convention must be stated when quoting an order.
The group group order group center is
quotient group by the center is quantum error-correcting
codes (Gottesman 1997, Nielsen and Chuang 2000).
The one-qubit Pauli group group order
16. It is isomorphic to the central product of the dihedral
group cyclic group subgroup isomorphic
to the quaternion group Cayley graph of Pauli
matrices is the Möbius-Kantor graph
(Bavuma et al. 2024).
See also Cayley Graph ,
Möbius-Kantor Graph ,
Pauli Matrices ,
Quantum
Error-Correcting Code ,
Quaternion Group ,
Qubit
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References Bavuma, Y.; D'Angeli, D.; Donno, A.; and Russo, F. G. "On an Infinite Family of Integral Cayley Graphs of Pauli Groups." J.
Algebra 659 , 148-182, 2024. https://doi.org/10.1016/j.jalgebra.2024.06.017 . Gottesman,
D. "Stabilizer Codes and Quantum Error Correction." Ph.D. thesis. Pasadena,
CA: California Institute of Technology, 28 May 1997. https://arxiv.org/abs/quant-ph/9705052 . Harper,
R.; Yu, W.; and Flammia, S. T. "Fast Estimation of Sparse Quantum Noise."
PRX Quantum 2 , 010322, 2021. https://doi.org/10.1103/PRXQuantum.2.010322 . Hashagen,
A. K.; Flammia, S. T.; Gross, D.; and Wallman, J. J. "Real Randomized
Benchmarking." Quantum 2 , 85, 2018. https://doi.org/10.22331/q-2018-08-22-85 . Knill,
O. "Remarks about the Möbius-Kantor Graph." 29 May 2026. https://arxiv.org/abs/2605.30799 . Nielsen,
M. and Chuang, I. Quantum
Computation and Quantum Information.
Cite this as:
Weisstein, Eric W. "Pauli Group." From
MathWorld https://mathworld.wolfram.com/PauliGroup.html
Subject classifications
, the 
-qubit Pauli group is the finite
matrix group of 
matrices consisting of scalar phases times Kronecker
products of the Pauli matrices. More explicitly,
on writing 
,
it is
is the 
identity matrix
and 
is the imaginary
unit. This convention includes all four scalar phases. Hashagen et al. (2018)
call the subgroup generated by Kronecker
products of 
and 
the real Pauli group, while Harper
et al. (2021) call 
the Paulis modulo phase. These groups have different orders,
so the convention must be stated when quoting an order.
has group order 
. Its group center is
, and the quotient group by the center is 
. The Pauli group is fundamental to the stabilizer
formalism for quantum error-correcting
codes (Gottesman 1997, Nielsen and Chuang 2000).
, also denoted 
(and 
by Knill 2026), has group order
16. It is isomorphic to the central product of the dihedral
group 
and the cyclic group 
, and contains a subgroup isomorphic
to the quaternion group 
. The Cayley graph of 
generated by the three Pauli
matrices is the Möbius-Kantor graph
(Bavuma et al. 2024).
