Pauli group

The $n$-qubit Pauli group consists of tensor products of $I,X,Y,Z$ multiplied by phases $\pm 1$ or $\pm i$.

For tensor products of Pauli factors from $\{I,X,Y,Z\}$, let $k$ count the sites where both factors are nonidentity and different. Distinct nonidentity Pauli matrices anticommute, and equal or identity factors commute. Reversing the product site by site therefore gives $PQ=(-1{)}^{k}QP$. Thus two Pauli strings commute exactly when $k$ is even. The criterion applies to the perfect-correlation observables in the GHZ theorem.

Store a Pauli group element with binary exponents $a_j,b_j$ and a phase exponent $s$ modulo four. This ordered $XZ$ convention uses $2n+2$ bits for $n$ qubits. Since $XZ=-iY$, phase factors are essential when converting to Hermitian Pauli operators and evaluating measurement probabilities.

Modulo scalar phases, the n-qubit Pauli group is the additive space ${\mathbb{F}}_2^{2n}$, with label $(a,b)$ for $X^a{Z}^b$. Multiplication adds labels, and commutation contributes the sign $(-1{)}^{\omega }$. The displayed alternating form is nondegenerate. An isotropic subspace $L$ satisfies $L\subseteq L^{\perp }$, hence $\dim L\le n$; it labels commuting stabilizers.

A subspace $L$ is isotropic when the symplectic pairing vanishes between every pair of its vectors. Its binary Pauli labels therefore commute. Nondegeneracy identifies the ambient space with its dual, so the map to $L^{\ast }$ is surjective and $\dim L^{\perp }=2n-\dim L$. Since $L\subseteq L^{\perp }$, it follows that $\dim L\le n$. This explains the maximum number of independent generators of a stabilizer group.

Conjugation by a Controlled-Z gate leaves the $a$ bits unchanged and updates the displayed $b$ bits and phase, using the old bits. The phase occurs when the extra $Z$ from one line is moved past $X$ on the other line. These constant-size updates support Heisenberg propagation of a Pauli observable through a Clifford circuit.

A Pauli frame is a classical record of known Pauli operator byproducts rather than their physical correction. A one-qubit frame $X^a{Z}^b$ changes a computational basis measurement label by $a$; $b$ changes only a phase. Conjugation by Clifford gates maps Pauli operators to Pauli operators, so the frame can be updated by classical binary arithmetic throughout a Clifford circuit.

An $n$-qubit Pauli operator is a tensor product of single-qubit Pauli matrices, possibly multiplied by a phase. Hermitian Pauli operators have eigenvalues $+1$ and $-1$.

Every linear map from a qubit into the tensor product of that qubit and an environment can be expanded in the Pauli matrix orthonormal basis of operators. If $V|0⟩=|0⟩|e_{00}⟩+|1⟩|e_{01}⟩$ and $V|1⟩=|0⟩|e_{10}⟩+|1⟩|e_{11}⟩$, then the four coefficient vectors areThese environment vectors need not be orthogonal, so this is an operator expansion rather than a claim that the quantum channel is a classical Pauli channel. Physical isometries additionally preserve the inner products of the two input basis states.

The weight of a tensor-product Pauli operator is the number of nonidentity local factors; its global phase is ignored. Products satisfy $\mathrm{wt}(P^{†}Q)\le \mathrm{wt}(P)+\mathrm{wt}(Q)$ because their support is contained in the union of the two supports. The quantum code distance uses this weight to measure the smallest undetectable nontrivial action on logical information.

The tensor products of single-qubit Pauli matrices form an orthogonal basis of the operator space on $n$ qubits, so every linear error operator has a Pauli expansion. Using $X^a{Z}^b$, with scalar phases absorbed into the coefficients, its computational basis action is $E|x⟩={\sum }_{a,b}{c}_{a,b}(-1{)}^{b\cdot x}|x+a⟩$, with binary arithmetic. This describes coherent errors as well as individual Pauli errors; it does not assume that the noise is a probabilistic Pauli channel. A physical noise map has Kraus operators of this form when the usual quantum channel description applies, for instance when the initial system and environment are uncorrelated.

A Pauli gate applies a Pauli matrix as a unitary operator to a qubit. The Pauli X gate, Pauli Y gate and Pauli Z gate are common correction operations in quantum teleportation.

A Pauli correlator is the expectation value of a tensor product of Pauli matrices, such as $⟨X\otimes Z⟩$. It measures a specified correlation between qubit observables.

A stabilizer group is an abelian subgroup of the Pauli group that does not contain $-I$. Its simultaneous positive eigenspace is its stabilizer subspace.

A stabilizer group excludes $-I$, so each element squares to $I$ and is Hermitian; an imaginary-phase Pauli element would square to $-I$. A nonidentity stabilizer contains at least one nonidentity local Pauli matrix and has zero trace, since the trace of a tensor product is the product of local traces. Its eigenvalues are therefore $\pm 1$ with equal multiplicities. This trace fact also gives $\dim {\mathcal{X}}_S=2^n/|S|$ from the stabilizer-projector formula.

A set of stabilizer generators is an independent commuting family of Hermitian Pauli operators whose products form a stabilizer group.

A stabilizer state on $n$ qubits is the unique simultaneous positive eigenstate of a stabilizer group with $n$ independent generators.

Stabilizer-state preparation starts from a computational-basis state and applies a Clifford circuit, producing a stabilizer state.

A stabilizer tableau represents each Pauli generator by a binary row $[x_1\cdots x_n\mid z_1\cdots z_n]$, with one additional bit when its sign is retained. Clifford gates update these rows by arithmetic over ${\mathbb{F}}_2$.

For an undirected graph $G=(V,E)$, the graph state is

Let $1$ be a leaf with neighbour $2$ and let the graph on the other vertices be generated by the Controlled-Z gate product $E_{\mathrm{rest}}$. If the leaf has Pauli Z gate frame $Z_1^r$, measuring it at angle $\alpha$ with result $s$ applies one-bit teleportation and leavesAny known Pauli Z gate frames on other vertices remain as additional factors before those vertices' input states. The identity $J(\alpha )Z^r=X^{r}J(\alpha )$ combines the leaf byproduct with the measurement result.

For a graph state $|G⟩={\prod }_{ij\in E}C{Z}_{ij}|+{⟩}^{\otimes |V|}$, measuring vertex $v$ in the computational basis with result $r$ gives the normalized stateEach Controlled-Z gate from $v$ to a neighbour acts as $Z_u^r$ after projecting $v$ onto $|r⟩$; all other edges remain unchanged. The projection contributes $1/\sqrt{2}$, so either result has probability $1/2$. For a four-cycle, deleting one vertex leaves a path and adds byproducts on its two endpoints.

The graph-state stabilizer generator at vertex $v$ is $S_v=X_v{\prod }_{u\in N(v)}{Z}_u$. Two such generators commute because adjacent vertices contribute two Pauli anticommutations and nonadjacent vertices contribute none.

Local complementation at a vertex $v$ toggles every edge between neighbors of $v$. It is implemented on graph states, up to global phase, by the local Clifford operator

The stabilizer subspace of a commuting family $S$ of Hermitian Pauli operators isIf there are $r$ independent generators on $n$ qubits, its dimension is $2^{n-r}$.

For a stabilizer group $G=⟨g_1,\dots ,g_l⟩$, the orthogonal projector onto its stabilizer subspace is

A Clifford gate is a unitary operation that normalizes the Pauli group: conjugating any Pauli operator by it produces another Pauli operator. Hadamard, phase, and controlled-$Z$ gates generate the Clifford operations.

A Clifford circuit is composed entirely of Clifford gates. It maps Pauli operators to Pauli operators in the Heisenberg picture and stabilizer states to stabilizer states in the Schrödinger picture.

The Gottesman--Knill theorem gives a classical polynomial-time simulation of quantum computations built from stabilizer-state preparation, Clifford gates, and adaptive Measurements of Pauli observables.

The extended Gottesman--Knill theorem allows arbitrary product-state inputs to a unitary Clifford circuit. Computational-basis output is weakly simulable for any number of measured qubits and strongly simulable for logarithmically many measured qubits.

A Clifford frame records a known Clifford transformation classically instead of applying it physically. Updating the frame changes the Pauli observables used for later measurements while preserving the represented quantum computation.

A local Clifford operation is a tensor product of one-qubit Clifford gates. On graph states, local Clifford operations implement graph transformations such as local complementation.

A strong classical simulation computes a quantum circuit's specified output probabilities in classical polynomial time to the requested polynomial precision. This is stronger than weak simulation, which only samples from the output distribution.

To compute a single-qubit output probability of a Clifford circuit, propagate its measured Pauli observable backward through the circuit. Every conjugation remains a tensor-product Pauli, whose expectation on a product state factors into efficiently computable one-qubit expectations.

For the binary phase representation of a Pauli string, these local rules implement $P\mapsto U^{†}PU$ for the Hadamard gate and the phase gate $S=\mathrm{diag}(1,i)$. In particular $S^{†}XS=-iXZ$, so the phase update differs in sign from forward conjugation. Use the old input bits, and reduce $s$ modulo four.

A weak classical simulation samples in classical polynomial time from the output distribution of a quantum circuit, exactly or within a prescribed small total-variation distance.