Pauli Group and Stabilizers

A stabilizer calculation begins with an elementary but unforgiving task: take a finite ordered list of signed Pauli strings and decide what its products and common eigenspace actually mean. This page develops a complete small-list audit. It keeps every phase, tests Hermiticity and compatibility, separates dependence from inconsistency, predicts the stabilized dimension, and turns commutation signs into an algebraic error signature. Stabilizer Formalism is the canonical full treatment of the scalable theory, including binary symplectic methods, projectors, normalizers, logical cosets, Clifford propagation, and tableaus. The present bridge stops before those general tools and before any claim about error correctability or a physical syndrome-extraction circuit.

Required background. Pauli Matrices supplies the standard Hermitian $Y=\sigma_y$ convention, one-qubit products, commutators, and tensor-product Pauli strings used in every direct audit here.

Helpful background. Why Quantum Error Correction Is Possible separates an algebraic error label from a correctability theorem. Multi-Qubit Gates supplies ordered tensor factors and outcome-resolved parity-measurement instruments; no measurement circuit is required for the algebraic audits on this page.

Signed Pauli Strings as Stabilizer Constraints

The word Pauli is used for several related objects. They must not be silently identified. A tensor word such as $X_1Z_3$ records a phase-free operator pattern, while closure under multiplication requires scalar phases. A check, in turn, must be a Hermitian observable with eigenvalues $\pm1$, so only a restricted part of the full group can be used directly. Finally, a list of checks becomes a stabilizer constraint only after its collective consistency has been audited.

Object	Typical notation	Allowed phase or relation	Role in this page
Phase-free Pauli string	$P_1\otimes\cdots\otimes P_n$	no recorded overall phase	labels tensor support but is not multiplication-closed
Phase-rich Pauli group	$i^\ell P_1\otimes\cdots\otimes P_n$	$\ell\in\{0,1,2,3\}$	supplies a closed multiplication law
Signed Hermitian check	$g=\pm P$	$g=g^\dagger$ and $g^2=I$	imposes the meaningful equation $g
Commuting generator list	$g_1,\ldots,g_m$	pairwise compatible signed checks	proposes a joint $+1$ sector, subject to further tests

This hierarchy is the first guardrail. Dropping phases may be harmless when one asks only whether two strings commute, but it is unsafe when one closes a generated group or assigns the sign of a constraint.

The phase-rich Pauli group

We use the standard Hermitian physics matrices

$$X= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}, \qquad Y= \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}, \qquad Z= \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}.$$

With this convention, $X$, $Y$, and $Z$ are Hermitian, square to $I$, and have eigenvalues $\pm1$. The full $n$-qubit Pauli group is

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

The four scalar phases are not decoration. For example, $XY=iZ$, so the phase-free set $\{I,X,Y,Z\}$ is not closed. Tensor products inherit the same issue: multiplying two apparently unsigned strings can produce an overall minus sign after two local factors of $i$ combine. Calderbank, Rains, Shor, and Sloane’s 1997 orthogonal-geometry formulation makes this commutation structure central, while their 1998 $\mathrm{GF}(4)$ formulation gives another efficient representation of Pauli labels. Those general encodings belong downstream; direct multiplication is more transparent for the finite lists considered here.

Preskill’s current Chapter 7 notes use the real anti-Hermitian symbol $Y=ZX=i\sigma_y$. That is a legitimate convention, but it is not the convention on this page. Here $Y=\sigma_y$ is Hermitian, so $ZX=iY$ and $XZ=-iY$. Every phase formula quoted or derived here has already been translated to this Hermitian convention.

Hermitian checks and meaningful signs

An operator used as a binary check must be measurable as a Hermitian observable. Thus a phase-rich element $iP$ or $-iP$, with $P$ Hermitian, is anti-Hermitian and cannot itself be a $\pm1$-valued check. The candidates are signed Hermitian strings $g=\pm P$. For either sign, $g^\dagger=g$ and $g^2=I$.

The sign of $g$ changes the constraint even though it does not change the tensor letters. If $P|\psi\rangle=|\psi\rangle$, then $(-P)|\psi\rangle=-|\psi\rangle$ rather than $|\psi\rangle$. Equivalently, the $+1$ sector of $-P$ is the $-1$ sector of $P$. A phase-free notation that turns $P$ and $-P$ into the same object therefore loses physical information needed to identify the proposed stabilized sector.

Gottesman’s 1996 code construction and 1997 thesis established signed commuting Pauli constraints as an economical language for quantum codes. The immediate lesson does not require a code construction: a signed list is a set of simultaneous eigenvalue equations. Before using it, one must ask whether those equations have any solution and how large the solution space is.

Phase-Safe Products and Commutation

Single-qubit multiplication supplies the only local rules needed for a direct audit:

$$XY=iZ, \qquad YZ=iX, \qquad ZX=iY,$$

with the sign-reversed opposite orders,

$$YX=-iZ, \qquad ZY=-iX, \qquad XZ=-iY.$$

To multiply strings, fix the tensor order, multiply at each position in that order, and multiply all scalar phases. Operators on different tensor factors commute. For example,

$$(X_1Y_2)(Z_1X_2) =(XZ)_1(YX)_2 =(-iY_1)(-iZ_2) =-Y_1Z_2.$$

Erasing the two factors of $-i$ would give the wrong sign. A leading sign must also be carried through the entire product. Finite audits are easiest to verify when each local product and the accumulated scalar are written explicitly.

Count local anticommutations

At one qubit, two Pauli letters commute if either is $I$ or if they are equal. Two distinct nonidentity letters anticommute. Let $P$ and $Q$ be phase-free $n$-qubit strings, and let $\nu(P,Q)$ count the positions where both local factors are nonidentity and different. Moving every local factor of $Q$ past the corresponding factor of $P$ contributes $-1$ at exactly those positions. Factors at distinct positions commute, hence

$$PQ=(-1)^{\nu(P,Q)}QP.$$

Only the parity of $\nu$ matters: an even number of local minus signs multiplies to $+1$, and an odd number multiplies to $-1$. Thus $P$ and $Q$ commute exactly when $\nu(P,Q)$ is even. Their overall scalar signs do not affect this commutation test, although those signs remain essential when products are closed.

Derive the overlap-parity rule

The rule can also be read directly from support overlap. Mark every tensor position at which the pair of letters is one of $X/Y$, $Y/Z$, or $Z/X$ in either order. Ignore positions containing $I$ and positions with identical letters. The parity of the marked positions is the commutation sign.

For instance, $X_1X_2X_3$ and $Z_1Z_2$ disagree nontrivially at qubits 1 and 2, so $\nu=2$ and the strings commute. By contrast, $X_1X_2$ and $Z_2Z_3$ have one such position, so they anticommute. This method determines commutation without calculating the product’s phase, but it does not determine whether the product is $+R$, $-R$, $+iR$, or $-iR$. A complete audit therefore uses overlap parity for compatibility and phase-safe multiplication for closure.

There is an immediate consistency consequence. If Hermitian checks $g$ and $h$ anticommute and a nonzero vector obeyed $g|\psi\rangle=h|\psi\rangle=|\psi\rangle$, then

$$gh|\psi\rangle=|\psi\rangle \quad\text{but}\quad gh|\psi\rangle=-hg|\psi\rangle=-|\psi\rangle,$$

a contradiction. Pairwise commutation is therefore necessary for a simultaneous $+1$ sector. It is not by itself sufficient.

A Consistency Audit for Candidate Generators

Given signed strings $g_1,\ldots,g_m$, audit them in an order that prevents an early shortcut from licensing a later claim. First verify that each proposed check is Hermitian. Then test every pair for commutation. Next close the generated subgroup far enough to exclude $-I$. Finally determine an independent generating rank rather than counting list entries. Gottesman’s 2026 draft develops this stabilizer logic systematically; the compact version here is designed for lists small enough to inspect by hand.

Audit item	Algebraic test	Failure meaning	Licensed consequence after a pass
Hermiticity	verify $g_j^\dagger=g_j$ and $g_j^2=I$	the item is not a binary Hermitian check	each listed equation has a $\pm1$ observable meaning
Pairwise commutation	require $g_jg_k=g_kg_j$ for every pair	no nonzero simultaneous eigenvector for the conflicting pair	simultaneous eigensectors are algebraically possible
Exclusion of $-I$	inspect all generated products and require $-I\notin\langle g_j\rangle$	the proposed $+1$ equations imply $-	\psi\rangle=
Independence	remove every generator expressible as a product of the others	raw list length overcounts constraints	independent rank $r$ determines group size and sector dimension

Each row answers a different question. In particular, dependence is not failure: it means that a listed equation repeats information already implied by the others.

Pairwise compatibility

For a short list, record $\nu(g_j,g_k)$ for each unordered pair. Odd parity is fatal to a common eigenbasis for that pair. Even parity says only that the pair is compatible; it does not certify the whole list. Because signed Hermitian strings that commute have Hermitian products, a passing pairwise test also ensures that closure does not unexpectedly produce anti-Hermitian group elements.

The distinction between pairwise and collective consistency matters. Three commuting signed checks can have products whose unsigned letters cancel while their accumulated sign is negative. The checks then pass every pairwise commutation test but generate $-I$, making the all-plus equations contradictory. This failure is invisible if one tracks only phase-free strings.

The minus-identity obstruction

Suppose a product of listed generators equals $-I$. Any vector stabilized by every generator would also be stabilized by their product, so it would satisfy

$$|\psi\rangle=(-I)|\psi\rangle=-|\psi\rangle,$$

and hence $|\psi\rangle=0$. Therefore a stabilizer group must exclude $-I$. The element $+I$ is harmless and necessarily belongs to every group. A nonempty generator product equal to $+I$ indicates redundancy, not inconsistency.

For a genuinely small list, explicit closure is the safest test: begin with $I$, multiply by each generator, retain the signs, and stop when no new element appears. Pairwise commutation makes the order immaterial but does not excuse dropping phases. If closure has many elements, that is precisely the point at which the binary formalism becomes preferable.

Independence and stabilized dimension

A commuting list is independent when no nonempty product of its members equals $I$. If $g_3=g_1g_2$, for example, then $g_3$ is redundant: removing it leaves the generated group and every joint eigenspace unchanged. If instead $g_1g_2g_3=-I$, the list is inconsistent rather than merely dependent. The sign separates the two cases.

For a finite hand audit, independence can be tested by labeling each subset with a bit string $\mathbf a=(a_1,\ldots,a_m)$ and multiplying

$$g(\mathbf a)=g_1^{a_1}\cdots g_m^{a_m}.$$

If two different bit strings yield the same signed operator, multiplying one result by the other gives a nonempty relation equal to $I$; one listed constraint is therefore implied by the rest. If two subsets yield opposite signed versions of the same phase-free string, their quotient is $-I$, exposing inconsistency instead. This comparison explains why phase-aware closure, independence, and the minus-identity test are closely related but cannot be collapsed into an unsigned rank calculation. Once a maximal independent subset is found, every discarded generator should be reconstructed explicitly as a signed product of that subset. That reconstruction is an auditable certificate of redundancy.

Let $r$ be the size of an independent generating list after redundancies are removed. For a valid rank-$r$ stabilizer group on $n$ qubits,

$$|S|=2^r, \qquad \dim\mathcal C=2^{n-r}.$$

Here $\mathcal C$ is the common $+1$ sector. We use this standard result without reproducing the canonical projector-and-trace proof. The raw number $m$ of supplied generators must never replace $r$: appending a product of existing generators increases $m$ but changes neither $S$ nor $\dim\mathcal C$.

The rank formula also clarifies what a sign choice does. Replacing an independent $g_j$ by $-g_j$ selects the opposite eigenvalue sector but does not alter the rank or the sector dimension, provided the entire signed list remains consistent. Appending a redundant generator cannot halve the dimension again, because its $+1$ equation already follows from the independent equations. The factor $2^{-r}$ therefore counts independent binary constraints, not lines in a presentation. This is why an audit reports both the supplied length $m$ and the reduced rank $r$ whenever they differ.

Dimension counting licenses only a statement about a joint eigenspace. It does not show that the sector corrects a chosen set of errors, determine its logical operators, or establish a distance. Knill and Laflamme’s 1997 criterion is the appropriate correctability theorem, and it requires information beyond the validity of the check group.

Error Signatures Before Syndrome Extraction

Once a valid commuting group is known, a Pauli error can be classified by whether it commutes or anticommutes with each check. This is an algebraic classification, independent of how any observable might later be measured. It is often called a syndrome in stabilizer discussions, but commutation-sign signature is the safer phrase here because no extraction procedure has been specified.

The commutation character

For a Pauli error $E$ and a stabilizer element $g$, define

$$gE=\chi_E(g)Eg, \qquad \chi_E(g)\in\{+1,-1\}.$$

The value is $+1$ for commutation and $-1$ for anticommutation. For commuting group elements $g$ and $h$,

$$\begin{aligned} (gh)E &=g(hE) =\chi_E(h)gEh\\ &=\chi_E(h)\chi_E(g)Egh, \end{aligned}$$

so

$$\chi_E(gh)=\chi_E(g)\chi_E(h).$$

Thus $\chi_E$ is a multiplicative character of the stabilizer group. This derivation is the finite bridge from a generator list to a consistent signature: generator signs determine the value on every group product. With ordered independent generators one may encode the same data as bits

$$s_j(E)=\frac{1-\chi_E(g_j)}{2}\in\{0,1\},$$

where $0$ means commute and $1$ means anticommute.

For independent generators, any assignment of $r$ signs extends uniquely to all $2^r$ group elements by multiplication. For a redundant presentation the displayed generator signs cannot be treated as freely specifiable coordinates. If $g_3=g_1g_2$, then every Pauli error must obey $\chi_E(g_3)=\chi_E(g_1)\chi_E(g_2)$. A purported ledger that violates this relation is internally inconsistent even if its entries were written as individual plus or minus symbols. Checking the character law against every known generator relation is therefore a useful finite validation step.

Notice also that replacing a check $g$ by $-g$ does not change $\chi_E(g)$, because an overall scalar commutes with $E$. It does change which physical eigenspace is called the all-plus sector. The signed group specifies the sector; the commutation character specifies how an error moves among its eigenvalue labels. Keeping these roles separate prevents an unsigned signature calculation from silently changing the stabilized space.

If $|\psi\rangle\in\mathcal C$, then $g|\psi\rangle=|\psi\rangle$ and

$$gE|\psi\rangle=\chi_E(g)E|\psi\rangle.$$

The error therefore moves a stabilized vector into the joint eigensector labeled by the character $\chi_E$. That is the exact conclusion supported by the algebra.

Signatures label sectors, not physical errors

Different Pauli errors can have the same signature. If $E$ and $F$ have identical characters, then $E^\dagger F$ commutes with every check; this relation does not force $E=F$. Even the all-plus signature is shared by every group element and by any Pauli operator in the group’s centralizer. Some such operators act trivially on $\mathcal C$, while others may act nontrivially within it.

Consequently a signature generally labels an equivalence class of physical errors and a joint eigensector, not a unique fault location or mechanism. Deciding which commuting operators are stabilizers and which represent logical cosets requires the normalizer quotient of the full formalism. Deciding whether equal-signature errors are jointly correctable requires the Knill–Laflamme conditions. Neither decision follows from the signature alone.

Algebra is not an extraction circuit

The relation $gE=\chi_E(g)Eg$ predicts how an ideal Pauli error changes a check eigenvalue. It does not tell an experimentalist how to couple an ancilla, in which order to apply gates, how to assign measurement outcomes, or how faults propagate through that circuit. Those are properties of an outcome-resolved measurement instrument and, later, of fault-tolerant syndrome extraction.

This separation prevents two common overclaims. First, calculating a signature does not demonstrate that it can be measured without disturbing encoded information. Second, a noiseless parity circuit does not by itself establish a fault-tolerant repeated-check protocol. The bridge remains entirely at the operator-algebra layer.

Worked Audit: Mixed-Support Checks

Consider the three-qubit signed checks

$$g_1=X_1X_2X_3, \qquad g_2=Z_1Z_2.$$

Both are Hermitian and square to $I$. They have two nonidentity, unequal overlaps, at qubits 1 and 2, so $\nu(g_1,g_2)=2$ and they commute. We now retain phases while closing their group and then compute a finite signature ledger.

Close the generated group

Using $XZ=-iY$ at the first two qubits,

$$\begin{aligned} g_1g_2 &=(X_1Z_1)(X_2Z_2)X_3\\ &=(-iY_1)(-iY_2)X_3\\ &=-Y_1Y_2X_3. \end{aligned}$$

The minus sign is essential. Since each generator squares to $I$ and they commute, no further distinct products occur:

$$\langle g_1,g_2\rangle = \{I,\;X_1X_2X_3,\;Z_1Z_2,\;-Y_1Y_2X_3\}.$$

The closure contains four Hermitian elements and not $-I$. Neither generator equals $I$, and their product is neither $I$ nor $-I$, so they are independent. Therefore $n=3$ and $r=2$, giving

$$n=3, \qquad r=2, \qquad \dim\mathcal C=2.$$

This completes the consistency and dimension audit. It does not yet say what information the two-dimensional sector encodes or which errors are correctable.

Build the signature ledger

For each listed error, overlap parity determines $\chi_E(g_1)$ and $\chi_E(g_2)$. The row for $g_1$ also checks the character logic: group elements commute with every element of this abelian group and therefore have the all-plus signature.

$E$	$\chi_E(g_1)$	$\chi_E(g_2)$	Algebraic reading
$I$	$+1$	$+1$	remains in the all-plus sector
$g_1$	$+1$	$+1$	stabilizer element with the all-plus signature
$X_1$	$+1$	$-1$	flips only the $g_2$ eigenvalue
$X_2$	$+1$	$-1$	aliases $X_1$ in this ledger
$X_3$	$+1$	$+1$	commutes with both checks
$Z_1$	$-1$	$+1$	flips only the $g_1$ eigenvalue
$Z_2$	$-1$	$+1$	aliases $Z_1$ in this ledger
$Z_3$	$-1$	$+1$	aliases $Z_1$ and $Z_2$
$Y_1$	$-1$	$-1$	flips both check eigenvalues
$Y_2$	$-1$	$-1$	aliases $Y_1$ in this ledger

The aliases are not calculation defects. There are four possible sign pairs for two independent checks but many Pauli errors. A sign pair can therefore distinguish joint eigensectors without identifying a unique operator.

The table can be cross-checked without repeating every overlap count. Since $Y_1$ differs from $X_1Z_1$ only by a scalar phase, its character is the pointwise product of the $X_1$ and $Z_1$ characters: $(+1,-1)(-1,+1)=(-1,-1)$. The same calculation gives the $Y_2$ row. Likewise, the alias relation between $X_1$ and $X_2$ implies that $X_1^\dagger X_2=X_1X_2$ has the all-plus signature. These are consequences of the character law, so they test the ledger as a coherent group assignment rather than as ten unrelated sign lookups.

State only the licensed conclusion

The audit establishes a consistent rank-two commuting group and a two-dimensional all-plus sector. It also predicts the eigensector reached by each listed Pauli error. In particular, $X_1$ and $X_2$ are aliases; $Z_1$, $Z_2$, and $Z_3$ are aliases; and $Y_1$ and $Y_2$ are aliases. The operator $X_3$ is undetected by these checks because it commutes with both.

The identity, $g_1$, and $X_3$ all have the all-plus signature, but the signature alone does not establish the same action on $\mathcal C$. The first two do act identically on stabilized vectors because $g_1|\psi\rangle=|\psi\rangle$; $X_3$ need not. Determining whether an all-plus operator belongs to the stabilizer or represents a nontrivial logical coset is canonical-formalism work.

Nothing in this ledger proves a quantum error-correcting capability. In particular, equal signatures may be acceptable for a degenerate correctable family, fatal for a proposed recovery, or associated with distinct logical actions. That judgment depends on products $E_a^\dagger E_b$ and the chosen stabilized sector, not merely on individual sign columns. This finite object should therefore not be called a three-qubit error-correcting code.

Nor does the dimension count select a preferred basis for the sector. A two-dimensional joint eigenspace can carry one qubit’s worth of amplitudes, but calling an operator a logical $X$ or logical $Z$ requires choosing representatives, checking their action modulo the stabilizer, and verifying their mutual algebra. The present audit supplies the raw all-plus sector and centralizer clues only. It deliberately leaves basis choice, logical-coset identification, distance, recovery, and experimental realization unstated.

Where the Full Stabilizer Formalism Begins

Direct audits expose the meaning of every sign and assumption, which is valuable at small $n$. They do not scale well. A list with $r$ independent generators has $2^r$ group elements, so explicit closure becomes exponentially cumbersome even though the generator description remains compact. The canonical formalism replaces repeated tensor multiplication with binary linear algebra and proves the general dimension, normalizer, and update results once.

Exit capability

After this page, a reader should be able to take any modest ordered list of signed Pauli strings and perform six actions: verify the $Y$ convention; multiply strings with their full phases; test Hermiticity and every commutation pair; distinguish $-I$ inconsistency from $+I$ redundancy; compute independent rank and stabilized dimension; and derive error signatures as group characters.

The exit capability is deliberately bounded. A passing audit says that the proposed joint $+1$ sector exists with the stated dimension. It does not identify logical operators, prove distance, select a recovery, validate a decoder, or specify a check-measurement circuit. Those conclusions require additional canonical objects and physical assumptions.

Canonical-owner handoffs

The next question determines the appropriate owner. Links identify the two implemented handoffs used directly from this bridge; other titles name either an established canonical page or a deliberately planned route without duplicating their material here.

Question or object	Canonical owner	Boundary preserved
Binary symplectic form, general projector, normalizer, logical cosets, Clifford propagation, tableaus, and Gottesman–Knill methods	Stabilizer Formalism	general and scalable stabilizer theory
Whether a specified error family is exactly or approximately correctable	Why Quantum Error Correction Is Possible	Knill–Laflamme conditions, degeneracy, and distance basics
Outcome-resolved parity measurements and ancilla circuits	Multi-Qubit Gates	measurement instruments and circuit realization
Cross-layer record from noise assumptions to recovery claims	Quantum Error Correction and Fault Tolerance	protection-claim escalation and chapter navigation
Physical extraction, repeated rounds, ancilla faults, and detector records	Syndrome Measurement	code-independent circuit and detector-record owner; this page retains the phase-safe algebraic signature
Worked Bell, GHZ, cluster, and graph-state constructions	Stabilizer States Preview	state examples rather than the readiness audit

The division is substantive rather than stylistic. A phase-safe algebraic list can be perfectly valid while failing to protect against the error family of interest. Conversely, a correctability theorem can be satisfied without dictating a fault-tolerant laboratory schedule. Each layer needs its own hypotheses and evidence.

Common Pitfalls

Treating phase-free strings as a group. The tensor letters are useful labels, but $XY=iZ$ leaves the phase-free set. Restore $\pm1$ and $\pm i$ before closing products or deciding whether $-I$ appears.

Using $\pm iP$ as a check. With the Hermitian $Y=\sigma_y$ convention, $\pm iP$ is anti-Hermitian when $P$ is Hermitian. A binary check must instead be a signed Hermitian string $\pm P$.

Stopping after pairwise commutation. Commuting generators can still multiply to $-I$. Pairwise compatibility must be followed by a collective phase-aware closure or an equivalent independence test.

Calling every dependence a contradiction. A product equal to $+I$ reveals a redundant generator and lowers the independent rank relative to the list length. A product equal to $-I$ makes the all-plus constraints inconsistent. The two signs lead to different verdicts.

Reading a signature as a unique fault address. The worked ledger contains several aliases, and the all-plus signature contains both stabilizers and potentially nontrivial commuting operators. A character labels an eigensector, not a unique microscopic error.

Promoting a signature to correctability. A valid group and a computed sign pattern do not imply the Knill–Laflamme conditions for a selected error set. Correctability compares pairs of errors on the protected sector.

Promoting algebra to a measurement protocol. A commutation sign predicts an ideal eigenvalue change but says nothing about ancilla preparation, gate ordering, outcome assignment, repeated rounds, or fault propagation.

Assuming every commuting operator is trivial. An operator that commutes with all checks preserves $\mathcal C$, but it may act nontrivially within that sector. Separating stabilizers from logical operators requires the normalizer and its cosets.

Exercises

Exercise 1: Multiply a Signed Pauli String

Using the Hermitian $Y=\sigma_y$ convention, calculate

$$A=(-X_1Y_2)(Z_1X_2) \qquad\text{and}\qquad B=(X_1I_2)(Y_1Z_2).$$

State whether each result is Hermitian. Identify the phase error made by replacing every local product with only its resulting Pauli letter.

Solution

For $A$, keep the leading minus sign and use $XZ=-iY$ and $YX=-iZ$:

$$A=-(-iY_1)(-iZ_2) =-(-i)^2Y_1Z_2 =Y_1Z_2.$$

The two factors of $-i$ multiply to $-1$, which cancels the supplied minus sign. Thus $A$ is Hermitian. For $B$,

$$B=(XY)_1Z_2=iZ_1Z_2.$$

This element is anti-Hermitian and is not a binary check. Letter-only multiplication would report $Y_1Z_2$ for $A$ without proving its sign and would incorrectly report the Hermitian string $Z_1Z_2$ for $B$, losing the decisive factor $i$.

Exercise 2: Test Overlap Parity

Let $P=X_1Y_2Z_4$ and $Q=Z_1Y_2X_3X_4$ on four qubits. Compute $\nu(P,Q)$ and decide whether $P$ and $Q$ commute. Then replace $Q$ by $Q'=Z_1X_2X_3X_4$ and repeat.

Solution

For $P$ and $Q$, qubit 1 contributes one anticommutation because $X$ and $Z$ differ, qubit 2 contributes none because both letters are $Y$, qubit 3 contributes none because $P$ has $I$, and qubit 4 contributes one because $Z$ and $X$ differ. Hence $\nu(P,Q)=2$ and

$$PQ=(-1)^2QP=QP.$$

For $Q'$, qubits 1, 2, and 4 each contain distinct nonidentity letters. Thus $\nu(P,Q')=3$ and

$$PQ'=(-1)^3Q'P=-Q'P.$$

The first pair commutes and the second anticommutes. No overall phase calculation is needed for this binary decision.

Exercise 3: Separate Fatal and Redundant Checks

Compare the two signed two-qubit lists

$$L_-=(X_1X_2,\;Z_1Z_2,\;-Y_1Y_2), \qquad L_+=(X_1X_2,\;Z_1Z_2,\;+Y_1Y_2).$$

Both lists are pairwise commuting. Decide which third entry is redundant and which list is inconsistent. Give the independent rank and stabilized dimension wherever those quantities exist.

Solution

The product of the first two checks is

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

In $L_-$, the third check is exactly the product of the first two. It is redundant, so the raw list length is three but the independent rank is $r=2$. The group excludes $-I$, and for $n=2$ its all-plus sector has dimension $2^{2-2}=1$.

In $L_+$, multiplying all three listed checks gives $(-Y_1Y_2)(+Y_1Y_2)=-I$. The list is therefore inconsistent and has no nonzero common $+1$ vector. The phase sign distinguishes harmless redundancy from a fatal obstruction.

Exercise 4: Replace Generators Without Changing the Sector

For the worked group generated by $g_1=X_1X_2X_3$ and $g_2=Z_1Z_2$, replace $g_2$ with $g'_2=g_1g_2=-Y_1Y_2X_3$. Show directly that $(g_1,g'_2)$ generates the same group, has the same independent rank, and stabilizes the same sector.

Solution

The new generator $g'_2=g_1g_2$ belongs to the original group, so every product of $g_1$ and $g'_2$ belongs to $\langle g_1,g_2\rangle$. Conversely,

$$g_1g'_2=g_1(g_1g_2)=g_1^2g_2=g_2,$$

where commutation and $g_1^2=I$ were used. Thus each old generator is generated by the new pair, and the groups coincide. Neither new generator is $I$, and their product is $g_2\ne\pm I$, so the pair remains independent with $r=2$ and $\dim\mathcal C=2$.

If a vector has eigenvalue $+1$ under $g_1$ and $g_2$, it has eigenvalue $+1$ under their product $g'_2$. Conversely, eigenvalue $+1$ under $g_1$ and $g'_2$ implies eigenvalue $+1$ under $g_1g'_2=g_2$. The stabilized sectors are identical.

Exercise 5: Find Signature Aliases

For the worked generators $g_1=X_1X_2X_3$ and $g_2=Z_1Z_2$, compute the signatures of $X_1$, $X_2$, $X_1X_2$, and $X_3$. Identify an alias pair and two distinct all-plus operators. Explain why these facts do not establish whether the four errors form a correctable set.

Solution

Both $X_1$ and $X_2$ commute with $g_1$ and anticommute with $g_2$, so each has signature $(+1,-1)$. They are aliases. Character multiplication gives

$$\chi_{X_1X_2}(g_j) =\chi_{X_1}(g_j)\chi_{X_2}(g_j),$$

so $X_1X_2$ has $(+1,+1)$. Directly, its two overlaps with $g_2$ also give even parity. The operator $X_3$ commutes with $g_1$ because its only nonidentity letter equals the third $X$, and it has disjoint support from $g_2$, so it too has $(+1,+1)$.

Thus $X_1,X_2$ are an alias pair, while $X_1X_2$ and $X_3$ are distinct undetected Paulis. A signature says which joint eigensector is reached; it does not determine how $E_a^\dagger E_b$ acts within $\mathcal C$. The Knill–Laflamme matrix elements, not the sign ledger alone, decide joint correctability.

Exercise 6: Complete a Finite Generator Audit

Audit the three-qubit list

$$h_1=X_1X_2, \qquad h_2=Z_1Z_2, \qquad h_3=Z_3.$$

Check Hermiticity, pairwise commutation, independence, and exclusion of $-I$; state the group size and stabilized dimension; and compute the signature of $E=X_1Y_3$. State exactly what the completed audit does and does not prove.

Solution

Each $h_j$ is a Hermitian signed Pauli string and squares to $I$. The first pair has two distinct nonidentity overlaps and therefore commutes; $h_3$ has disjoint support from both $h_1$ and $h_2$. The product of the first pair is

$$h_1h_2=(-iY_1)(-iY_2)=-Y_1Y_2.$$

The four products generated by $h_1,h_2$ are $I,h_1,h_2,-Y_1Y_2$, all acting trivially on qubit 3. Multiplying any of them by $h_3=Z_3$ gives a distinct string with nonidentity third support. Hence no nonempty product is $I$ or $-I$: the list is independent, has rank $r=3$, and generates $2^3=8$ elements without the minus identity. With $n=3$,

$$\dim\mathcal C=2^{3-3}=1.$$

For $E=X_1Y_3$, the $X_1$ factor commutes with $h_1$ and anticommutes with $h_2$, while $Y_3$ anticommutes with $h_3$. Therefore

$$(\chi_E(h_1),\chi_E(h_2),\chi_E(h_3))=(+1,-1,-1).$$

The audit proves that a one-dimensional common $+1$ sector exists and that $E$ maps it to the stated joint eigensector. It does not supply an extraction circuit, identify a physical fault, or certify correction of any error family.

References

• A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum Error Correction and Orthogonal Geometry,” Physical Review Letters 78, 405–408 (1997), DOI.
• A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum Error Correction Via Codes Over GF(4),” IEEE Transactions on Information Theory 44, 1369–1387 (1998), DOI, arXiv.
• D. Gottesman, “A Class of Quantum Error-Correcting Codes Saturating the Quantum Hamming Bound,” Physical Review A 54, 1862–1868 (1996), DOI.
• D. Gottesman, Stabilizer Codes and Quantum Error Correction, Ph.D. thesis, California Institute of Technology (1997), repository DOI, arXiv, author errata.
• D. Gottesman, Surviving as a Quantum Computer in a Classical World, 2026 draft, official PDF.
• E. Knill and R. Laflamme, “Theory of Quantum Error-Correcting Codes,” Physical Review A 55, 900–911 (1997), DOI.
• J. Preskill, Quantum Error Correction, Chapter 7, updated March 2026, official PDF.