Pauli group

The Pauli group on ${\displaystyle n}$ qubits, ${\displaystyle G_{n}}$, is the group generated by the operators described above applied to each of ${\displaystyle n}$ qubits in the tensor product Hilbert space ${\displaystyle (\mathbb {C} ^{2})^{\otimes n}}$. That is,

$${\displaystyle G_{n}=\langle P_{1}\otimes \cdots \otimes P_{n}:P_{i}\in \{I,X,Y,Z\}\rangle =\{c\cdot P_{1}\otimes \cdots \otimes P_{n}:c\in \{\pm 1,\pm i\},P_{i}\in \{I,X,Y,Z\}\}.}$$

The order of ${\displaystyle G_{n}}$ is ${\displaystyle 4\cdot 4^{n}}$ since a scalar ${\displaystyle \pm 1}$ or ${\displaystyle \pm i}$ factor in any tensor position can be moved to any other position.

An ${\displaystyle n}$-qubit Pauli operator that only acts on a single qubit is often denoted as a single Pauli letter with an integer subscript. For example, in a system with 3 qubits,

${\displaystyle X_{1}\equiv X\otimes I\otimes I,\qquad Z_{2}\equiv I\otimes Z\otimes I.}$

Multi-qubit Pauli operators can be written as products of single-qubit Paulis on disjoint qubits. Alternatively, when it is clear from context, the tensor product symbol ${\displaystyle \otimes }$ can be omitted, i.e. unsubscripted Pauli matrices written consecutively represents tensor product rather than matrix product. For example:

${\displaystyle XZI\equiv X_{1}Z_{2}=X\otimes Z\otimes I.}$

Operators in ${\displaystyle G_{n}}$ can also be represented as ${\displaystyle 2^{n}\times 2^{n}}$ matrices. An operator ${\displaystyle P=c\cdot P_{1}\otimes \cdots \otimes P_{n}}$ always has two distinct eigenvalues, either ${\displaystyle \pm 1}$ or ${\displaystyle \pm i}$ depending on whether the scalar factor ${\displaystyle c}$ is ${\displaystyle \pm 1}$ or ${\displaystyle \pm i}$. An operator with eigenvalues ${\displaystyle \pm 1}$ is Hermitian, and one with eigenvalues ${\displaystyle \pm i}$ is anti-Hermitian. In either case, a set of ${\displaystyle 2^{n}}$ eigenvectors of ${\displaystyle P}$ can be constructed by taking tensor products of eigenvectors of each ${\displaystyle P_{i}}$, with the eigenvalue being ${\displaystyle c}$ times the product of the eigenvalues of each factor.

Two operators in ${\displaystyle G_{n}}$ either commute or anti-commute, depending on whether the number of anti-commuting pairs of single-qubit Pauli operators at the same location is even or odd. For example, ${\displaystyle XXX}$ and ${\displaystyle ZZX}$ commute with each other since there are exactly two anti-commuting pairs (on qubits 1 and 2), but ${\displaystyle XXX}$ and ${\displaystyle ZZZ}$ anti-commute since there are three such pairs.

A simple but useful mapping ${\displaystyle N}$ exists between the binary vector space ⁠${\displaystyle (\mathbb {Z} _{2})^{2}}$⁠ and the set of Pauli matrices ⁠${\displaystyle \{I,X,Y,Z\}}$⁠:

⁠${\displaystyle 00\to I,\;01\to X,\;11\to Y,\;10\to Z.}$⁠

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 ⁠${\displaystyle I,X,Y,Z}$⁠ are regarded as representatives of equivalence classes in the quotient group ⁠${\displaystyle [G]=G/\{\pm 1,\pm i\}}$⁠ (where ⁠${\displaystyle G}$⁠ is the single-qubit Pauli group). For ⁠${\displaystyle P\in G}$⁠, denote the equivalence class represented by ⁠${\displaystyle P}$⁠ as

⁠${\displaystyle [P]=\{\beta P\mid \beta \in \{\pm 1,\pm i\}\}.}$⁠

Note that ⁠${\displaystyle [G]}$⁠ is a commutative group since two Pauli operators either commute or anti-commute, but ⁠${\displaystyle [-P]=[P]}$⁠.

The map ⁠${\displaystyle N}$⁠ now induces an isomorphism ⁠${\displaystyle [N]:(\mathbb {Z} _{2})^{2}\mapsto [G]}$⁠, i.e., addition of vectors in ⁠${\displaystyle (\mathbb {Z} _{2})^{2}}$⁠ is equivalent to multiplication of Pauli operators up to a global phase:

⁠${\displaystyle [N(u+v)]=[N(u)][N(v)].}$⁠

Furthermore, let ⁠${\displaystyle \odot }$⁠ denote the symplectic product between two elements ⁠${\displaystyle u,v\in (\mathbb {Z} _{2})^{2}}$⁠, where ⁠${\displaystyle u=z\vert x}$⁠ and ⁠${\displaystyle v=z'\vert x'}$⁠ (this notation represents binary string concatenation, e.g., ⁠${\displaystyle 01\equiv 0\vert 1}$⁠), ⁠${\displaystyle z,x,z',x'\in \mathbb {Z} _{2}}$⁠:

⁠${\displaystyle u\odot v\equiv zx'-xz'.}$⁠

Then the symplectic product ⁠${\displaystyle \odot }$⁠ gives the commutation relations of elements of ⁠${\displaystyle G}$⁠:

⁠${\displaystyle N(u)N(v)=(-1)^{(u\odot v)}N(v)N(u).}$⁠

The symplectic product and the mapping ⁠${\displaystyle N}$⁠ 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 ⁠${\displaystyle \mathbf {N}$ :(\mathbb {Z} _{2})^{2n}\mapsto G_{n}} ⁠ such that

⁠${\displaystyle \mathbf {N} (\mathbf {z} \vert \mathbf {x} )=N(z_{1}\vert x_{1})\otimes \cdots \otimes N(z_{n}\vert x_{n}).}$⁠

Similar to the single-qubit case, denoting the quotient group ⁠${\displaystyle G_{n}/\{\pm 1,\pm i\}}$⁠ as ⁠${\displaystyle [G_{n}]}$⁠, the map ⁠${\displaystyle [\mathbf {N} ]:(\mathbb {Z} _{2})^{2n}\mapsto [G_{n}]}$⁠ is an isomorphism:

⁠${\displaystyle [\mathbf {N} (\mathbf {u} +\mathbf {v} )]=[\mathbf {N} (\mathbf {u} )][\mathbf {N} (\mathbf {v} )].}$⁠

Furthermore, for ⁠${\displaystyle \mathbf {u} =\mathbf {z} \vert \mathbf {x} }$⁠ and ⁠${\displaystyle \mathbf {v} =\mathbf {z} '\vert \mathbf {x} '}$⁠, where ⁠${\displaystyle \mathbf {z} ,\mathbf {x} ,\mathbf {z} ',\mathbf {x} '\in (\mathbb {Z} _{2})^{n}}$⁠, define the symplectic product ⁠${\displaystyle \odot }$⁠ as

⁠${\displaystyle \mathbf {u} \odot \mathbf {v} \equiv \sum _{i=1}^{n}z_{i}x_{i}'-x_{i}z_{i}'=\sum _{i=1}^{n}u_{i}\odot v_{i},}$⁠

where ⁠${\displaystyle u_{i}=z_{i}\vert x_{i}}$⁠ and ⁠${\displaystyle v_{i}=z_{i}'\vert x_{i}'}$⁠. Then the symplectic product captures the commutation relations of any operators ⁠${\displaystyle \mathbf {N} (\mathbf {u} )}$⁠ and ⁠${\displaystyle \mathbf {N} (\mathbf {v} )}$⁠:

⁠${\displaystyle \mathbf {N} (\mathbf {u} )\mathbf {N} (\mathbf {v} )=(-1)^{(\mathbf {u} \odot \mathbf {v} )}\mathbf {N} (\mathbf {v} )\mathbf {N} (\mathbf {u} ).}$⁠

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.

• Nielsen, Michael A.; Chuang, Isaac L. (2000). Quantum Computation and Quantum Information. Cambridge; New York: Cambridge University Press. ISBN 978-0-521-63235-5. OCLC 43641333.