DefinitionDefinition: Literature-sourcedProof: 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.

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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.

Szegedy reflection walk and its two-reflection square

Definition

For the finite row-stochastic kernel and row-preparation map of Szegedy row-preparation map and discriminant, let $Π : = T T^{*}$ , the orthogonal projection onto $ran (T)$ , and choose the reflection sign

$R_{A} : = 2 Π - I .$

With $S$ the two-register swap, define the one-reflection Szegedy step and its square by

$U : = S R_{A}, W : = U^{2} = (S R_{A} S) R_{A} = (2 S Π S - I) (2 Π - I) .$

Products act right to left. This fixes $U$ and $W$ as different operators. Algebraically, if $U v = μ v$ , then $W v = μ^{2} v$ on the same vector. The other common reflection sign is $I - 2 Π = - R_{A}$ and is not interchangeable inside a controlled operation.

Depends on

Used by

MNRS finite reversible-walk finding Algorithm
Szegedy-walk marked-set detection Algorithm
Finite row-preparation access model for a Szegedy walk Definition
Two-state reversible walk and its endpoint phases Example
Szegedy reflections and walk operators are unitary Lemma
Szegedy walk invariant subspaces from discriminant eigenvectors Lemma
Exact row-preparation access count for one and two walk steps Theorem
Szegedy spectral mapping for one and two reflections Theorem

Dependency tree · two levels

15 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

Andrew M. Childs, Lecture Notes on Quantum Algorithms, §§18.2–18.3 (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, §8.2 (standard reference, not scraped)