DefinitionDefinition: AI-adaptedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Stabilizer syndromes and logical Pauli operators

Definition

Let $n \geq 1$ and let $S$ be a stabilizer subgroup generated by an ordered independent Hermitian list $g_{1}, \dots, g_{r}$ . For a phase-aware Pauli error $E$ , its syndrome is the bit string $s (E) \in F_{2}^{r}$ defined by

$E g_{j} = (- 1)^{s_{j} (E)} g_{j} E, s_{j} (E) : = ω (λ (E), λ (g_{j})) .$

Here $λ (P (k, x, z)) = (x, z)$ is the binary Pauli label and $ω ((x, z), (x^{'}, z^{'})) = x \cdot z^{'} + z \cdot x^{'}$ . The scalar center of $P_{n}$ is $Z_{n} = {i^{k} I : k \in Z / 4 Z}$ , and the normalizer is $N (S) = {P \in P_{n} : P g = g P for every g \in S}$ . The logical Pauli classes form the quotient group

$N (S) / (Z_{n} S) .$

A normalizer element is logically nontrivial exactly when it lies in $N (S) ∖ Z_{n} S$ . Multiplication by a scalar phase does not change a syndrome or logical class.

Facts & Assumptions

Given: The phase-aware Pauli group, its binary label, a stabilizer group $S$ , and an ordered Hermitian generating list.

[F1]

Binary Pauli label and quotient: $λ (P (k, x, z)) = (x, z)$ forgets only the scalar phase.

[F2]

Binary symplectic form: $ω ((x, z), (x^{'}, z^{'})) = x \cdot z^{'} + z \cdot x^{'}$ .

[F3]

Pauli commutator is symplectic pairing: the Pauli commutator sign is $(- 1)^{ω (λ (P), λ (Q))}$ .

Verification

1.1givenF2F3

Applying [F3] to $E$ and $g_{j}$ gives $E g_{j} = (- 1)^{ω (λ (E), λ (g_{j}))} g_{j} E$ . The exponent is in $F_{2}$ , so it determines exactly one bit, and this verifies the stated sign-to-bit convention.

2.1F1F3step 1.1algebra

If $E^{'} = i^{a} E$ for a scalar phase $i^{a}$ , then $E^{'} g_{j} = i^{a} E g_{j}$ and $g_{j} E^{'} = i^{a} g_{j} E$ ; the same commutation sign results. Equivalently, $λ (E^{'}) = λ (E)$ by [F1]. Hence the syndrome depends only on the Pauli coset modulo scalar phase.

3.1givenF1∎

If $E \in Z_{n} S$ , write $E = z s$ with $z \in Z_{n}$ and $s \in S$ . The scalar $z$ commutes with every generator, and $s$ commutes with every generator because $S$ is abelian, so $E \in N (S)$ and its logical quotient class is the identity. Conversely the denominator of the displayed quotient is exactly the scalar phases times stabilizers; thus a normalizer element outside $Z_{n} S$ represents a nonidentity logical Pauli class.

Depends on

Used by

Noncontractible toric logical loops Definition
Pauli-path energy barrier Definition
Stabilizer distance Definition
Toric e and m syndromes Definition
CSS encoding from nested binary codes Example
Steane seven-qubit CSS code Example
Minimum-Weight Recovery Failure Yields a Half-Error Crossing Path Lemma
CSS syndrome correction Theorem
Dimension of the logical Pauli quotient Theorem
Exact indistinguishability in a contractible disk Theorem
Planar Surface-Code Memory Bound Under IID Z Noise Theorem
Square planar surface-code distance and path count Theorem
Stabilizer distance detection and correction Theorem
The periodic toric code encodes two qubits Theorem
Toric code distance is L Theorem
Two anticommuting toric logical Pauli pairs Theorem

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John Preskill, Quantum Computation, Chapter 7: Quantum Error Correction (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, §20.1 (standard reference, not scraped)