DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-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.

Character annihilator of a subgroup

Definition

Let $G$ be a finite abelian group with character dual $\hat{G}$ (Dual group of a finite abelian group). For a subgroup $H \leq G$ (Subgroup), define its character annihilator by $H^{⊥} : = {χ \in \hat{G} : χ (h) = 1 for every h \in H} .$ For any subset $K \subseteq \hat{G}$ (in particular, for any subgroup once $\hat{G}$ has been proved to be a group), define $K^{⊥} : = {g \in G : χ (g) = 1 for every χ \in K} .$ The pairing is evaluation, and the same convention is used in both directions. lem-qa-finite-abelian-annihilators-are-subgroups proves that these sets are subgroups when the input is a subgroup.

Depends on

Used by

Finite abelian hidden-subgroup common-kernel reconstruction Algorithm
Finite abelian hidden-subgroup promise and output encoding Definition
Common-kernel subgroup recovery in a product of cyclic groups Example
Finite abelian Fourier and convolution examples Example
Fourier sampling for a cyclic subgroup Example
A Fourier sample cuts a larger common kernel Lemma
Annihilators of subgroups are subgroups Lemma
Chernoff repetition bound for common-kernel recovery Lemma
Fourier annihilator equation for cyclic discrete logarithm Lemma
Double annihilator returns the subgroup Theorem
Fourier samples are uniform in the annihilator Theorem
Fourier transform of a subgroup coset indicator Theorem
Subgroup-annihilator cardinality Theorem

Dependency tree · two levels

7 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

Ronald de Wolf, Quantum Computing: Lecture Notes, version 6 (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms (standard reference, not scraped)
John Preskill, Quantum Information, Chapter 6: Quantum Algorithms (standard reference, not scraped)