In physics, quantum information and group theory, the Pauli group is a group formed by tensor products of Pauli matrices, including the identity.[1] The single-qubit Pauli group is a 16-element matrix group, consisting of the 4 Pauli matrices each with 4 possible phase factors. The -qubit Pauli group is a
4n+1
In quantum information theory, Pauli groups are important because they are the basis for stabilizer formalism, a widely-used framework for constructing and describing quantum error correction codes using sets of commuting Pauli operators. Stabilizer codes are formed from commuting subgroups of the Pauli group.[3] [4]
I
X=\sigma1=\begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}, Y=\sigma2=\begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}, Z=\sigma3= \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}
\pm1
\pmi
G \stackrel{def
As an abstract group,
G \congC4\circD4
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
\sigma1\sigma2\sigma3=iI
The Pauli algebra is the algebra of 2 x 2 complex matrices M(2, C) with matrix addition and matrix multiplication. It has a long history beginning with the biquaternions introduced by W. R. Hamilton in his Lectures on Quaternions (1853). The representation with matrices was noted by L. E. Dickson in 1914.[6] Publications by Pauli eventually led to the eponym now in use. Basis elements of the algebra generate the Pauli group.
The Pauli group on
n
Gn
n
(C2) ⊗
The order of
Gn
4 ⋅ 4n
\pm1
\pmi
An
n
X1\equivX ⊗ I ⊗ I, Z2\equivI ⊗ Z ⊗ I.
⊗
XZI\equivX1Z2=X ⊗ Z ⊗ I.
Operators in
Gn
2n x 2n
P=c ⋅ P1 ⊗ … ⊗ Pn
\pm1
\pmi
c
\pm1
\pmi
\pm1
\pmi
2n
P
Pi
c
Two operators in
Gn
XXX
ZZX
XXX
ZZZ
A simple but useful mapping
N
This mapping allows a multi-qubit Pauli operator to be represented as a binary vectors with a phase factor, and operations on these operators to be defined as binary operations rather than matrix operations.
Some useful properties of this mapping becomes evident when the phaseless Pauli operators are regarded as representatives of equivalence classes in the quotient group (where is the single-qubit Pauli group). For, denote the equivalence class represented by as
Note that is a commutative group since two Pauli operators either commute or anti-commute, but .
The map now induces an isomorphism, i.e., addition of vectors in is equivalent to multiplication of Pauli operators up to a global phase:
Furthermore, let denote the symplectic product between two elements, where and (this notation represents binary string concatenation, e.g.,), :
Then the symplectic product gives the commutation relations of elements of :
The symplectic product and the mapping thus give a useful way to phrase Pauli relations in terms of binary algebra.
The above definitions can be straightforwardly extended to multiple qubits, defining a mapping such that
Similar to the single-qubit case, denoting the quotient group as, the map is an isomorphism:
Furthermore, for and, where, define the symplectic product as
where and . Then the symplectic product captures the commutation relations of any operators and :
The above binary representation and symplectic algebra are especially useful in making the relation between classical linear error correction and quantum stabilizer codes more explicit. In the language of symplectic vector spaces, a symplectic subspace corresponds to a direct sum of Pauli algebras (i.e., encoded qubits), while an isotropic subspace corresponds to a set of stabilizers.
