LemmaStatement: Literature-sourcedProof: 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.

Normalization is stable under a small flagged-vector error

Statement

Let $v$ be an ideal flagged vector with $∥ v ∥ \geq 1 / (2 κ)$ . If $0 < ϵ \leq 1$ and either $∥ u - v ∥ \leq ϵ / (8 κ)$ or $∥ u - v ∥ \leq (9 ϵ / 100) ∥ v ∥$ , then $u \neq 0$ and $∥ u / ∥ u ∥ - v / ∥ v ∥ ∥ \leq ϵ$ .

Facts & Assumptions

Given: The two vectors and the stated lower and error bounds.

[F1]

Ideal inverse amplitude and HHL postselection probability: the ideal well vector has norm at least $1 / (2 κ)$ .

[F2]

The inner-product norm is definite, homogeneous, and satisfies the triangle inequality: $∥ x + y ∥ \leq ∥ x ∥ + ∥ y ∥$ ; applying it to $u = (u - v) + v$ and $v = (v - u) + u$ gives $∣ ∥ u ∥ - ∥ v ∥ ∣ \leq ∥ u - v ∥$ .

Proof

1.1givenF1F2algebra

Under the absolute-error alternative, the reverse triangle inequality from [F2] gives $∥ u ∥ \geq ∥ v ∥ - ∥ u - v ∥ \geq (1 / 2 - 1 / 8) / κ = 3 / (8 κ) > 0$ . Under the relative alternative, $∥ u ∥ \geq (1 - 9 ϵ / 100) ∥ v ∥ > 0$ because $9 ϵ / 100 < 1$ . Thus both normalizations exist.

2.1givenF1F2step 1.1algebra∎

Add and subtract $u / ∥ v ∥$ and use [F2]: $∥ \frac{u}{∥ u ∥} - \frac{v}{∥ v ∥} ∥ \leq \frac{∣ ∥ v ∥ - ∥ u ∥ ∣}{∥ v ∥} + \frac{∥ u - v ∥}{∥ v ∥} \leq \frac{2 ∥ u - v ∥}{∥ v ∥} .$ The absolute alternative makes this at most $2 (ϵ / (8 κ)) / (1 / (2 κ)) = ϵ / 2$ . The relative alternative makes it at most $18 ϵ / 100 < ϵ$ .

Depends on

Used by

OP5 HHL returns the QLSP state within the requested error Theorem

Dependency tree · two levels

18 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

Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd, Quantum algorithm for linear systems of equations (standard reference, not scraped)
Andrew M. Childs, Robin Kothari, and Rolando D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms (standard reference, not scraped)