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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 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.

The Pauli group includes scalar phases $\mathcal{Z}=\{\pm I,\pm iI\}$. For a nonzero stabilizer code, its Abelian subgroup must satisfy $-I\notin S$. Otherwise a common positive eigenvector would satisfy $-|\psi ⟩=|\psi ⟩$ and vanish; for example $S=\{I,-I\}$ is Abelian but yields no nonzero stabilizer code. We impose this standard admissibility condition.

Every element of such a stabilizer group is Hermitian and squares to $I$: an imaginary-phase Pauli with square $-I$ is excluded. The associated stabilizer code and its orthogonal projector areThe group multiplication proves $P^2=P$, Hermiticity gives $P^{†}=P$, and its range is exactly the common positive eigenspace. If $S$ has $r$ independent generators, $|S|=2^r$. Every nonidentity element has trace zero, while $\mathrm{tr}I=2^n$, soThus it encodes $k=n-r$ logical qubits.

For the symplectic representation of a stabilizer code, label $i^t{X}^a{Z}^b$ by $(a,b)\in {\mathbb{F}}_2^{2n}$, ignoring scalar phase. Products add labels, andThis is the nondegenerate alternating form $\omega ((a,b),(a^{\prime },b^{\prime }))=a\cdot b^{\prime }+b\cdot a^{\prime }$ on the binary symplectic space of Pauli labels. The image $L$ of $S$ is an r-dimensional isotropic subspace of a binary symplectic space, since the generators commute and no nontrivial scalar lies in $S$. Consequently $L\subseteq L^{\perp }$ and $r\le n$. A binary generator matrix $(A\mid B)$ satisfies $A{B}^T+B{A}^T=0$. Conversely an isotropic subspace can be lifted by choosing commuting Hermitian generators with independent labels; their products cannot equal $-I$, and the orthogonal projector formula supplies the stabilizer code. The labels of the centralizer of a stabilizer group are $L^{\perp }$, with nontrivial logical actions represented by $L^{\perp }\setminus L$.

For an example, the three-qubit bit-flip repetition code has $S=⟨Z_1{Z}_2,Z_2{Z}_3⟩$ and ${\mathcal{C}}_S=\mathrm{span}\{|000⟩,|111⟩\}$. Its two binary rows have zero X part and Z parts $110,011$. The errors $I,X_1,X_2,X_3$ have respective error syndromes $00,10,11,01$, so all single bit flips can be corrected. This is not a stabilizer code correcting every single-qubit error: $Z_1$ acts as a nontrivial logical phase error.

Let $C(S)$ denote the Pauli centralizer. The exact Pauli error criterion for a stabilizer code isEquivalently, no difference of error labels lies in $L^{\perp }\setminus L$. The scalar phases in $\mathcal{Z}S$ are essential: an error product $-I$ is harmless even though it is not in $S$.

We prove the criterion. For $T=E_a^{†}{E}_b$ outside $C(S)$, some $s\in S$ anticommutes with $T$. Since $sP=Ps=P$, we have $PTP=PsTsP=-PTP$, hence $PTP=0$. If $T=\lambda s\in \mathcal{Z}S$, then $PTP=\lambda P$. If $T\in C(S)\setminus \mathcal{Z}S$, it restricts to a unitary operator on the stabilizer code, butno $sT$ is scalar. A scalar restriction would have scalar coefficient zero by this trace equation, contradicting unitarity. Thus this third case is not scalar.

To justify why scalar compressions are exactly what correction requires, suppose one recovery corrects every $E_a$. Dilate that recovery to an isometry $W$ including its auxiliary system. Correct recovery of every pure stabilizer code state gives $W{E}_a|\psi ⟩=|\psi ⟩\otimes |{\eta }_a⟩$. The auxiliary state is independent of $\psi$: linearity on basis states and their superpositions forces their auxiliary vectors to coincide. Preservation of inner products then giveswhich is the Knill--Laflamme condition $P{E}_a^{†}{E}_{b}P=c_{ab}P$. The three cases above prove necessity of the boxed criterion.

For sufficiency, put errors in the same class when $E_a^{†}{E}_b\in \mathcal{Z}S$. They act identically on the stabilizer code up to scalar phase, so give the same error-image subspace. Under the criterion, different classes give mutually orthogonal subspaces $E_a{\mathcal{C}}_S$, because their cross-compressions vanish. Measure which of these subspaces is occupied and apply the inverse of a representative error. This returns every stabilizer code state unchanged; extend the recovery on the orthogonal complement by preparing any fixed stabilizer code state. Linearity also corrects any valid noise channel whose Kraus operators are linear combinations of the specified errors: within each syndrome all amplitudes multiply the same logical state, and trace preservation normalizes the recovered state. This proves sufficiency, including degenerate error classes.

The physical space of $n$ qubits is ${\mathcal{H}}_n=({\mathbb{C}}^2{)}^{\otimes n}$. An error operator is a linear operator on this space, for example one Kraus operator of a noise quantum channel. The usual channel description assumes initially uncorrelated system and environment, with the environment subsequently ignored; more general initial correlations need not define a channel on arbitrary system inputs. Finite dimension and the tensor decomposition into identifiable qubits allow an exact Pauli expansion of a quantum error. No independence between errors on different qubits is assumed by this expansion.

The $4^n$ tensor products of $I,X,Y,Z$ form an orthogonal basis of the operator space under the Hilbert-Schmidt inner product. Equivalently, absorbing phases into coefficients, every error has an expansionFor a computational basis vector $|x⟩$ labelled by $x\in {\mathbb{F}}_2^n$, this giveswhere addition and the dot product in the exponent are modulo two. Thus $X$ changes a bit, $Z$ changes its phase, and $Y$ performs both up to a scalar phase. A general error is a coherent linear combination of such actions; it need not be a classical random choice of Pauli operators. A restriction to at most $t$ faulty qubits is an additional locality assumption, expressed by the span of Pauli operators of weight at most $t$.

The weight of a Pauli operator is the number of tensor factors different from $I$, ignoring global phase. Let $P$ be the orthogonal projector onto a $2^k$-dimensional code $\mathcal{X}$. Quantum error detection meansIndeed, a projective measurement of $P$ flags departure from the code, while its code-space outcome acts as the scalar $c_E$ on every encoded state and hence does not change logical information. Harmless scalar actions are included in this criterion. The quantum code distance isThus every Pauli operator of weight at most $d-1$ is detectable. The condition is linear in $E$, so it holds for any error in their span. This proves detection of arbitrary errors affecting at most $d-1$ qubits, including coherent errors.

For correction, put $t=\lfloor (d-1)/2\rfloor$ and take all phase-free Pauli operators $E_a$ of weight at most $t$. Their pairwise products satisfyThese are the Knill--Laflamme conditions. Here is their recovery construction, rather than merely using the criterion's name. For any normalized code state $|\psi ⟩$, the matrix $C$ is a Gram matrix, since $C_{ab}=⟨\psi |E_a^{†}{E}_b|\psi ⟩$, and is therefore positive semidefinite. Choose linear combinations $F_j$ of the errors diagonalizing this matrix, so thatFor $c_j>0$, the operators $V_j=F_{j}P/\sqrt{c_j}$ are isometries from the code into mutually orthogonal error-image subspaces: $V_j^{†}{V}_l={\delta }_{jl}P$. Measure the projectors $Q_j=V_j{V}_j^{†}$ and, on outcome $j$, apply $V_j^{†}$. Complete this recovery arbitrarily on the remaining orthogonal subspace to make a trace-preserving quantum channel.

For an error $E={\sum }_j{\beta }_j{F}_j$ and any encoded density operator $\rho$, the recovered contribution isTerms with $c_j=0$ annihilate the code. For a physical noise channel whose Kraus operators belong to this span, summing the displayed scalar over those operators gives one by trace preservation. The resulting recovered state is exactly $\rho$. This proves the constructive recovery from the Knill-Laflamme condition and the requested guaranteeThis argument allows degenerate codes; distinct physical errors may have identical actions on the code.

For the Shor code, define $|G_{\pm }⟩=(|000⟩\pm |111⟩)/\sqrt{2}$. Its two normalized basis codewords areSix stabilizer generators are $Z_1{Z}_2,Z_2{Z}_3,Z_4{Z}_5,Z_5{Z}_6,Z_7{Z}_8,Z_8{Z}_9$; the two phase checks areAll eight have eigenvalue $+1$ on both basis codewords. A phase error $Z_4$ commutes with the six $Z$ checks and anticommutes with both $g_7$ and $g_8$. Their projective measurements therefore returnThe two negative phase checks identify the middle three-qubit block. Apply the Pauli Z gate $Z_4$ to restore the state, since $Z_4^2=I$. This procedure acts identically on every logical superposition, and the checks reveal no logical amplitudes. The syndrome cannot distinguish $Z_4$ from $Z_5$ or $Z_6$, but all three have the same action on the code: their pairwise products lie in the stabilizer group. Correcting with any one of them is sufficient. Thus the degeneracy in the syndrome is harmless.

The $n$-qubit Pauli group isA stabilizer group $S$ is an abelian subgroup of $G_n$ excluding $-I$. Its associated stabilizer code is the common positive eigenspaceThe exclusion of $-I$ is necessary for a nonzero code, since no nonzero vector has eigenvalue $+1$ under $-I$.

Every $s\in S$ is a signed Hermitian tensor product of Pauli matrices. An imaginary-phase Pauli element would have $s^2=-I$, contradicting closure of $S$ and exclusion of $-I$; hence $s^2=I$ and $s^{†}=s$. If $s\ne I$, it cannot be a nontrivial scalar Pauli element, and at least one of its local factors is $X,Y$ or $Z$. Each of these has trace zero. Using $\mathrm{Tr}(A\otimes B)=\mathrm{Tr}(A)\mathrm{Tr}(B)$ gives $\mathrm{Tr}s=0$.

The spectral theorem for Hermitian operators and $s^2=I$ show that all eigenvalues are $+1$ or $-1$. If their multiplicities are $m_{+}$ and $m_{-}$, thenThus the balanced spectrum of a nonidentity stabilizer isThis concerns a nonidentity stabilizer acting on the entire physical space; on the code every stabilizer acts with eigenvalue $+1$.

Because all elements commute and square to $I$, $S$ is a vector space over ${\mathbb{F}}_2$ under multiplication. Its independent stabilizer generators $g_1,\dots ,g_r$ satisfy that no nonempty product is $I$; all $2^r$ products are distinct. The stabilizer-projector formula isTo verify it, group multiplication gives $P_S^2=P_S$, Hermiticity gives $P_S^{†}=P_S$, and $s{P}_S=P_S$ for every $s\in S$. Conversely every code vector is fixed by this average, so its range is exactly ${\mathcal{X}}_S$. All nonidentity stabilizers have trace zero by the preceding proof. ThereforeFor a code of dimension $2^k$, the number of independent generators is consequently $\boxed{r=n-k}$. One can list additional redundant generators, but an independent generating set has exactly this size.

For a correctable Pauli error $E_{\alpha }$, define the syndrome bit $s_j(\alpha )$ to be zero when $E_{\alpha }$ commutes with $g_j$ and one when it anticommutes. On an encoded state,Measure the commuting stabilizer generators; their eigenvalues reveal the error syndrome without revealing logical amplitudes. In a nondegenerate quantum code, the distinct specified correctable errors have orthogonal images of the code and distinct syndromes. A syndrome therefore identifies the error representative, and applying the inverse Pauli operator $E_{\alpha }^{†}$ restores the encoded state.

If an actual error operator is a coherent linear combination of the specified errors, its state is a sum of these orthogonal syndrome components. Syndrome measurement selects one component, followed by the same inverse operation. The linearity of coherent quantum error correction ensures the logical state is recovered in every outcome. The same procedure handles noise Kraus operators in that error span; it does not require a probabilistic Pauli-noise assumption.

A Pauli error commuting with every stabilizer has the all-zero syndrome and preserves the code space. If it is a stabilizer up to global phase, this silent action is harmless. If it lies in the centralizer of a stabilizer group but is not a stabilizer up to phase, it acts as a nontrivial logical operator, and syndrome measurements cannot detect it. More generally, a non-scalar component $P_{S}E{P}_S$ of an error is an undetectable logical action. This distinguishes harmful undetected errors from scalar actions that satisfy quantum error detection algebraically.

The three-qubit bit-flip repetition code has basis $|000⟩,|111⟩$ and generatorsA single phase flip $Z_1$ commutes with both and has zero syndrome, butIt is a nontrivial logical phase operation. In particular, $P_S{Z}_1{P}_S$ is not scalar, so the error set $\{I,Z_1\}$ violates the Knill--Laflamme condition and cannot be corrected. Thus the code corrects single bit flips but not single phase flips; its distance against arbitrary quantum errors is one.

The state $|001⟩$ has independent stabilizer generators $ZII$, $IZI$, and $-IIZ$. Its complete stabilizer group isFor $|{\Phi }^{+}⟩\otimes |+⟩$, take the generators $XXI$, $ZZI$, and $IIX$. Their products giveIn particular $(XX)(ZZ)=-YY$, which accounts for the two minus signs.

An $n$-qubit stabilizer state is the unique simultaneous $+1$ eigenstate of an abelian group $G$ of Pauli operators having $2^n$ elements and not containing $-I$. The group $G$ is the state's stabilizer group; equivalently, it is generated by $n$ independent commuting Hermitian Pauli operators.

Choose a Clifford operation $D$ with $D|0^n⟩=|\sigma ⟩$ and absorb $D$ into the circuit. Push each subsequent Clifford gate forward through the computation. A computational-basis measurement made after a Clifford prefix $U$ becomes a Pauli measurementon the initial state, because Clifford conjugation preserves the Pauli group. Adaptivity merely makes the next Pauli depend on earlier classical outcomes.

It remains to eliminate the $n$ stabilizer qubits. Maintain their current stabilizer group. For a Pauli $P$ to be measured, there are two cases.

maps the old stabilizer sector into the $\lambda$ eigenspace of $P$. Updating the Clifford frame by $V_{\lambda }$ removes this measurement while conjugating every later Pauli to another Pauli.

Iterating this procedure leaves an adaptive Pauli-based computation on $|\rho ⟩$. The same classical outcomes determine every adaptive choice and final output, so this gives a weak classical simulation. Every postselected $Z$ outcome becomes either a fixed classical $+1$ branch or a postselected $+1$ Pauli measurement, as required.

If $S|\alpha ⟩=|\alpha ⟩$ and $|\beta ⟩=U|\alpha ⟩$, thenThus conjugating the state conjugates its entire stabilizer group.

The triangle $G_B$ is obtained from the path $G_A$ by local complementation of a graph state at vertex $2$, which toggles the edge between its neighbors $1$ and $3$. The corresponding Local Clifford operation isDirect conjugation givesThese three commuting operators generate exactly the same stabilizer group as $S_1^B,S_2^B,S_3^B$. Thereforefor an irrelevant global phase $e^{i\varphi }$.

The commuting termsare independent stabilizer generators. Since $J>0$, every ground state has $S_j=+1$. The resulting two-dimensional stabilizer subspace isthe phase-flip repetition code. A convenient pair of logical Pauli operators isIndeed, they commute with every stabilizer, anticommute with each other, and are not stabilizers.

For phase-flip errors $E_A={\prod }_{j\in A}{Z}_j$ and $E_B={\prod }_{j\in B}{Z}_j$, the operator entering the Knill--Laflamme condition is $E_A^{†}{E}_B=E_{A△B}$ and has weight at most $2t$. Every nonempty proper product of $Z_j$ anticommutes with some $S_j$ unless it is the full logical operator $\overline{X}$. The Knill--Laflamme conditions therefore hold whenever $2t<n$, and fail once two allowed errors can differ by $\overline{X}$. Thus

Each physical $X_j$ commutes with all stabilizers, and $X_j{X}_1$ is a product of stabilizers. Hence every $X_j$ acts on the code as the undetectable logical operator $\overline{Z}$. The code cannot detect, and therefore cannot correct, even a single bit flip.

A stabilizer code is the common $+1$ eigenspace of a stabilizer group $S$, an abelian subgroup of the Pauli group excluding $-I$. Choose $r$ independent stabilizer generators $g_1,\dots ,g_r$ on $n$ physical qubits. They are commuting Hermitian operators, each squaring to $I$. The stabilizer subspace and its orthogonal projector areThe stabilizer-projector formula gives $\dim \mathcal{C}=2^{n-r}$, so $k=n-r$ logical qubits remain. Independent here means that no nonempty product of the chosen stabilizer generators is $I$. Excluding $-I$ is necessary: a vector cannot simultaneously be fixed by $I$ and $-I$ unless it is zero.

For the three-qubit bit-flip repetition code, take $g_1=Z_1{Z}_2$ and $g_2=Z_2{Z}_3$, with $Z_j$ the Pauli Z gate on site $j$ and identities elsewhere. The stabilizer group is $S=\{I,Z_1{Z}_2,Z_2{Z}_3,Z_1{Z}_3\}$. Both stabilizer generators have eigenvalue $+1$ exactly on the span of $|000⟩$ and $|111⟩$, giving $2^{3-2}=2$ dimensions for one logical qubit. A projective measurement of these checks distinguishes four orthogonal two-dimensional sectors rather than eight individual computational basis states.

A Pauli error that anticommutes with a stabilizer generator changes that check's sign, producing an error syndrome. A Pauli operator commuting with every check can still act on the logical qubits; the centralizer of a stabilizer group describes these possibilities. Thus the number of checks determines the encoded dimension, while the quantum code distance determines protection against arbitrary local errors. The five-qubit error correcting code, for instance, has four stabilizer generators and quantum code distance three; the three-qubit bit-flip repetition code has two checks but quantum code distance one because a single phase flip is unflagged. ^[ref].

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

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

A stabilizer group with $r$ independent generators maps injectively to an r-dimensional isotropic subspace of the binary symplectic space of Pauli labels. For generator rows $(A\mid B)$, commutation is $A{B}^T+B{A}^T=0$ over ${\mathbb{F}}_2$. Its stabilizer subspace has dimension $2^{n-r}$. The labels of the centralizer of a stabilizer group form $L^{\perp }$; labels in $L^{\perp }\setminus L$ act as nontrivial logical Pauli operators.