Stabilizer Identities

Conventions

The $n$-qubit Pauli group is

$$\mathcal P_n = \left\{ i^\ell P_1\otimes\cdots\otimes P_n: \ell\in\{0,1,2,3\}, \ P_j\in\{I,X,Y,Z\} \right\}.$$

A stabilizer constraint uses a Hermitian signed Pauli $g$:

$$g^\dagger=g, \qquad g^2=I, \qquad g\lvert\psi\rangle = \lvert\psi\rangle.$$

The sign is physical bookkeeping. The operators $g$ and $-g$ select opposite eigenspaces.

Binary vectors are written in the order

$$\mathbf v = (\mathbf x\mid\mathbf z) \in \mathbb F_2^{2n}.$$

All binary sums and products on this page are evaluated modulo $2$. Pauli phase or sign must be stored separately.

At a glance

Object	Formula or test
Hermitian Pauli representative	$P(\mathbf x,\mathbf z)=i^{\sum_jx_jz_j}X(\mathbf x)Z(\mathbf z)$
Symplectic product	$\omega(\mathbf v,\mathbf w)=\mathbf x\cdot\mathbf z'+\mathbf z\cdot\mathbf x'$
Commutation	$P(\mathbf v)P(\mathbf w)=(-1)^{\omega(\mathbf v,\mathbf w)}P(\mathbf w)P(\mathbf v)$
Check-matrix constraint	$H\Lambda H^{\mathsf T}=0$
Group order	$\lvert S\rvert=2^r$ for $r$ independent generators
Code projector	$P_S=2^{-r}\sum_{g\in S}g$
Code dimension	$\dim\mathcal C(S)=2^{n-r}$
Encoded qubits	$k=n-r$
Syndrome	$\mathbf s=H\Lambda\mathbf e^{\mathsf T}$
Logical Pauli classes	$N(S)/S$
Stabilizer-code distance	$d=\min_{Q\in N(S)\setminus S}\operatorname{wt}(Q)$
Clifford condition	$U\mathcal P_nU^\dagger=\mathcal P_n$

Binary Pauli representation

For binary strings $\mathbf x,\mathbf z\in\mathbb F_2^n$, define

$$X(\mathbf x) = \prod_{j=1}^n X_j^{x_j}, \qquad Z(\mathbf z) = \prod_{j=1}^n Z_j^{z_j}.$$

A convenient Hermitian representative is

$$P(\mathbf x,\mathbf z) = i^{\sum_jx_jz_j} X(\mathbf x)Z(\mathbf z).$$

On one qubit, the support dictionary is

$(x_j,z_j)$	Pauli factor
$(0,0)$	$I$
$(1,0)$	$X$
$(0,1)$	$Z$
$(1,1)$	$Y$

The vector records support and commutation, but not the complete signed operator. For example, $Y$ and $-Y$ have the same binary vector.

Symplectic commutation test

For

$$\mathbf v=(\mathbf x\mid\mathbf z), \qquad \mathbf w=(\mathbf x'\mid\mathbf z'),$$

define

$$\omega(\mathbf v,\mathbf w) = \mathbf x\cdot\mathbf z' + \mathbf z\cdot\mathbf x'.$$

Then

$$P(\mathbf v)P(\mathbf w) = (-1)^{\omega(\mathbf v,\mathbf w)} P(\mathbf w)P(\mathbf v).$$

Thus

$$\omega=0 \quad\Longleftrightarrow\quad [P(\mathbf v),P(\mathbf w)]=0,$$

and

$$\omega=1 \quad\Longleftrightarrow\quad \{P(\mathbf v),P(\mathbf w)\}=0.$$

With row vectors, introduce

$$\Lambda = \begin{pmatrix} 0&I_n\\ I_n&0 \end{pmatrix}.$$

The same test is

$$\omega(\mathbf v,\mathbf w) = \mathbf v\Lambda\mathbf w^{\mathsf T}.$$

Equivalently, two Pauli strings commute when the number of qubits on which they have different nonidentity factors is even.

Stabilizer group and check matrix

A qubit stabilizer group $S$ is an abelian subgroup of $\mathcal P_n$ with

$$-I\notin S.$$

Suppose

$$S = \langle g_1,\ldots,g_r \rangle$$

has $r$ independent commuting Hermitian generators. Independence means

$$g_1^{a_1}\cdots g_r^{a_r} = I \quad\Longrightarrow\quad a_1=\cdots=a_r=0$$

for $a_j\in\mathbb F_2$. Consequently,

$$\lvert S\rvert = 2^r.$$

Place the binary generator rows in

$$H = \left[ H_X\mid H_Z \right].$$

Pairwise commutation is equivalent to

$$H\Lambda H^{\mathsf T} = 0,$$

or

$$H_XH_Z^{\mathsf T} + H_ZH_X^{\mathsf T} = 0.$$

Count the binary rank of $H$, not the number of rows in a redundant generator list.

For a CSS code with separate $X$-type and $Z$-type check matrices, the cross-commutation condition reduces to

$$H_XH_Z^{\mathsf T} = 0.$$

Projector and code dimension

The projector onto the common $+1$ eigenspace is

$$\begin{aligned} P_S &= \prod_{j=1}^{r} \frac{I+g_j}{2} \\ &= \frac{1}{2^r} \sum_{g\in S}g. \end{aligned}$$

Only the identity Pauli has nonzero trace. Since

$$\operatorname{Tr}I = 2^n,$$

the code-space dimension is

$$\begin{aligned} \dim\mathcal C(S) &= \operatorname{Tr}P_S \\ &= 2^{n-r}. \end{aligned}$$

An $[[n,k]]$ stabilizer code therefore has

$$r=n-k$$

independent generators. The cases are:

Rank	Stabilized space
$r=n$	one-dimensional stabilizer state, $k=0$
$r<n$	code subspace encoding $k=n-r$ qubits
$r>n$	impossible for independent commuting qubit Paulis

For a stabilizer state,

$$\rho_S = \lvert\psi_S\rangle\langle\psi_S\rvert = \frac{1}{2^n} \sum_{g\in S}g.$$

For an $[[n,k]]$ code, the same Pauli sum is the maximally mixed encoded state:

$$\rho_{\mathrm{code}} = \frac{P_S}{2^k} = \frac{1}{2^n} \sum_{g\in S}g.$$

Pauli expectation values

For a stabilizer state and a Hermitian Pauli $Q$,

$$\langle Q\rangle = \begin{cases} +1, & Q\in S, \\ -1, & -Q\in S, \\ 0, & \text{otherwise}. \end{cases}$$

The zero case follows because some stabilizer generator anticommutes with $Q$. For a code state, every $g\in S$ still has expectation $+1$, but a Pauli commuting with all checks may be a logical observable whose expectation depends on the encoded state.

This gives a fast diagnostic: stabilizer membership predicts deterministic Pauli outcomes; anticommutation with a check predicts zero expectation.

Syndromes

Let an error Pauli $E$ have binary vector

$$\mathbf e = (\mathbf e_X\mid\mathbf e_Z).$$

Define syndrome bit $s_j$ by

$$s_j = \begin{cases} 0, & [E,g_j]=0, \\ 1, & \{E,g_j\}=0. \end{cases}$$

In matrix form,

$$\mathbf s = H\Lambda\mathbf e^{\mathsf T} = H_X\mathbf e_Z^{\mathsf T} + H_Z\mathbf e_X^{\mathsf T}.$$

After $E$ acts on an ideal code state, check $g_j$ has measured eigenvalue

$$(-1)^{s_j}.$$

The syndrome map is linear:

$$\mathbf s(\mathbf e+\mathbf f) = \mathbf s(\mathbf e) + \mathbf s(\mathbf f).$$

Two Pauli errors $E$ and $F$ have the same syndrome exactly when

$$E^\dagger F \in N(S)$$

up to phase. Equal syndrome does not imply equal logical action.

Logical Paulis and distance

Within the Pauli group, define

$$N(S) = \left\{ Q\in\mathcal P_n: Qg=gQ \text{ for every }g\in S \right\}.$$

Paulis fall into three operational classes:

Relation to $S$	Effect on the code
Anticommutes with some $g\in S$	leaves the code space and produces a nonzero syndrome
Belongs to $S$ up to phase	acts trivially on code states
Belongs to $N(S)\setminus S$	preserves the code but acts as a nontrivial logical Pauli

Logical Pauli equivalence classes form

$$N(S)/S.$$

Multiplying a representative by a stabilizer does not change its encoded action:

$$\overline Q \sim \overline Qg, \qquad g\in S.$$

The stabilizer-code distance is

$$d = \min_{Q\in N(S)\setminus S} \operatorname{wt}(Q),$$

with global phase ignored. A low-weight stabilizer does not lower $d$ because it acts trivially.

For a Pauli error set $\mathcal E$, exact correction is possible precisely when

$$E_a^\dagger E_b \notin N(S)\setminus S$$

for every $E_a,E_b\in\mathcal E$, up to phase. Their relative product may either be detectable or be a stabilizer, but it must not be an undetected logical Pauli.

Clifford conjugation

A unitary $U$ is Clifford when

$$U\mathcal P_nU^\dagger = \mathcal P_n.$$

If $\lvert\psi\rangle$ is stabilized by $S$, then

$$U\lvert\psi\rangle$$

is stabilized by

$$USU^\dagger = \left\{ UgU^\dagger:g\in S \right\}.$$

Useful one- and two-qubit propagation rules are:

Gate	Conjugation rules
$H_j$	$X_j\mapsto Z_j$, $Z_j\mapsto X_j$, $Y_j\mapsto-Y_j$
$S_j$	$X_j\mapsto Y_j$, $Y_j\mapsto-X_j$, $Z_j\mapsto Z_j$
$\operatorname{CNOT}_{c\to t}$	$X_c\mapsto X_cX_t$, $Z_t\mapsto Z_cZ_t$, while $Z_c$ and $X_t$ stay fixed
$\operatorname{CZ}_{a,b}$	$X_a\mapsto X_aZ_b$, $X_b\mapsto Z_aX_b$, while both $Z$ operators stay fixed

Signs must be propagated. Tracking only the binary support can produce the wrong eigenspace.

Ignoring phase, a Clifford acts by a binary matrix $F$:

$$\mathbf v \longmapsto \mathbf vF.$$

Preservation of Pauli commutation requires

$$F\Lambda F^{\mathsf T} = \Lambda.$$

The symplectic matrix $F$ is not the complete Clifford description; phase data are also required.

Pauli measurement update

Let $M$ be a Hermitian Pauli with outcome $m\in\{+1,-1\}$.

1. If $M\in S$, the outcome is deterministically $+1$.
2. If $-M\in S$, the outcome is deterministically $-1$.
3. If $M$ commutes with every stabilizer but $M\notin\pm S$, it acts as a logical Pauli and measuring it generally reveals logical information.
4. If $M$ anticommutes with a stabilizer generator, ideal outcomes are equiprobable.

For the fourth case, choose one anticommuting generator $g_j$. Multiply every other anticommuting generator by $g_j$, then replace $g_j$ with

$$mM.$$

The resulting commuting generators stabilize the postmeasurement state. Generator replacement describes the ideal state update; ancilla faults and repeated noisy syndrome extraction require a circuit-level analysis.

Canonical examples

Single-qubit states

The computational and Hadamard-basis states have stabilizers

$$\begin{aligned} \lvert0\rangle &: \langle Z\rangle, & \lvert1\rangle &: \langle -Z\rangle, \\ \lvert+\rangle &: \langle X\rangle, & \lvert-\rangle &: \langle -X\rangle. \end{aligned}$$

The signs distinguish orthogonal states with the same Pauli support.

Bell state

The Bell state

$$\lvert\Phi^+\rangle = \frac{ \lvert00\rangle+\lvert11\rangle }{ \sqrt2 }$$

is uniquely stabilized by

$$S_{\Phi^+} = \langle X_1X_2, Z_1Z_2 \rangle.$$

The projector identity gives

$$\lvert\Phi^+\rangle\langle\Phi^+\rvert = \frac14 \left( I + X_1X_2 - Y_1Y_2 + Z_1Z_2 \right).$$

The minus sign appears because

$$(X_1X_2)(Z_1Z_2) = -Y_1Y_2.$$

GHZ state

For

$$\lvert\mathrm{GHZ}_n\rangle = \frac{ \lvert0\rangle^{\otimes n} + \lvert1\rangle^{\otimes n} }{ \sqrt2 },$$

one independent generating set is

$$X_1X_2\cdots X_n, \qquad Z_1Z_2, \quad Z_2Z_3, \quad \ldots, \quad Z_{n-1}Z_n.$$

There are $n$ independent generators, so the stabilized space is one-dimensional.

Graph state

For a simple graph $G=(V,E)$, the graph-state generators are

$$K_v = X_v \prod_{u\in N(v)}Z_u, \qquad v\in V.$$

The $K_v$ commute and are independent. They define a unique stabilizer state. Local graph-state transformations and entanglement structure belong to the canonical graph-state page.

Compact audit

For a proposed stabilizer description, check:

1. qubit and syndrome-bit ordering;
2. generator signs;
3. pairwise symplectic products;
4. binary rank rather than row count;
5. exclusion of $-I$;
6. dimension $2^{n-r}$;
7. logical representatives modulo $S$;
8. phase updates under Clifford gates;
9. whether a zero-syndrome operator is a stabilizer or a logical Pauli;
10. whether an implementation claim includes a physical fault model.

Common mistakes

• Dropping Pauli phases and then treating the remaining set as a group.
• Treating $g$ and $-g$ as the same stabilizer constraint.
• Allowing noncommuting generators or including $-I$.
• Counting redundant rows as independent checks.
• Confusing a stabilizer state, with $k=0$, with a stabilizer code.
• Calling every zero-syndrome Pauli harmless.
• Assuming same-syndrome errors have the same logical action.
• Forgetting that $F\Lambda F^{\mathsf T}=\Lambda$ does not store Clifford phase data.
• Calling every physical Clifford a fault-tolerant logical gate.
• Applying binary qubit formulas unchanged to qudits, subsystem codes, or bosonic codes.

Exercises

1. Rank and code dimension

Five qubits have four listed commuting generators, but one generator is the product of the other three. What is the stabilized-space dimension?

Solution

The binary rank is $r=3$, not $4$. Therefore,

$$\dim\mathcal C = 2^{5-3} = 4.$$

The code encodes

$$k=5-3=2$$

logical qubits.

2. Symplectic test

Determine whether

$$P=X_1Z_2Y_3, \qquad Q=Z_1X_2$$

commute.

Solution

The binary vectors are

$$\begin{aligned} \mathbf v_P &= (1,0,1\mid0,1,1), \\ \mathbf v_Q &= (0,1,0\mid1,0,0). \end{aligned}$$

Their symplectic product is

$$\begin{aligned} \omega(P,Q) &= \mathbf x_P\cdot\mathbf z_Q + \mathbf z_P\cdot\mathbf x_Q \\ &= 1+1 \\ &= 0 \pmod2. \end{aligned}$$

They commute. Directly, the strings anticommute on qubits $1$ and $2$, so the two minus signs cancel.

3. Bell-projector sign

Starting from

$$S = \langle X_1X_2,Z_1Z_2\rangle,$$

derive the Pauli expansion of $P_S$.

Solution

The stabilizer group has four elements:

$$S = \left\{ I, X_1X_2, Z_1Z_2, (X_1X_2)(Z_1Z_2) \right\}.$$

Because $XZ=-iY$ on each qubit,

$$(X_1X_2)(Z_1Z_2) = (-iY_1)(-iY_2) = -Y_1Y_2.$$

Therefore,

$$P_S = \frac14 \left( I + X_1X_2 - Y_1Y_2 + Z_1Z_2 \right).$$

4. Syndrome versus logical action

Suppose two Pauli errors $E$ and $F$ have the same syndrome. What further test decides whether one recovery can correct both?

Solution

Equal syndrome implies

$$E^\dagger F \in N(S)$$

up to phase. If

$$E^\dagger F \in S,$$

the errors differ by a stabilizer and have the same logical action. One recovery can correct both.

If instead

$$E^\dagger F \in N(S)\setminus S,$$

the relative product is a nontrivial logical Pauli. The same syndrome-conditioned recovery cannot correct both without leaving a logical error for one case.

5. Clifford propagation

Start from $\lvert00\rangle$, stabilized by

$$\langle Z_1,Z_2\rangle.$$

Propagate the generators through $H_1$ followed by $\operatorname{CNOT}_{1\to2}$.

Solution

Hadamard maps $Z_1$ to $X_1$:

$$\langle Z_1,Z_2\rangle \longmapsto \langle X_1,Z_2\rangle.$$

CNOT maps control $X$ to $X_1X_2$ and target $Z$ to $Z_1Z_2$:

$$\langle X_1,Z_2\rangle \longmapsto \langle X_1X_2,Z_1Z_2\rangle.$$

The final state is $\lvert\Phi^+\rangle$.

Canonical links

• Stabilizer Formalism
• Why Quantum Error Correction Is Possible
• Stabilizer Simulation
• Stabilizer States Preview
• Graph States
• GHZ States
• Quantum Gates
• Pauli Matrix Table

References

• D. Gottesman, Stabilizer Codes and Quantum Error Correction, Ph.D. thesis, California Institute of Technology (1997), doi:10.7907/rzr7-dt72.
• D. Gottesman, “The Heisenberg representation of quantum computers”, in Group22: Proceedings of the XXII International Colloquium on Group Theoretical Methods in Physics (1999), arXiv:quant-ph/9807006.
• S. Aaronson and D. Gottesman, “Improved simulation of stabilizer circuits”, Physical Review A 70, 052328 (2004), doi:10.1103/PhysRevA.70.052328.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press (2010), Chapter 10.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press (2018), Sections 3.2 and 7.5.