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.

The QLSP target is nonzero

Statement

Under the hypotheses in Quantum linear-systems state-output task, $A^{+} b \neq 0$ , so the normalized target is well-defined.

Facts & Assumptions

Given: A finite-dimensional map $A$ and $b \in ran (A) ∖ {0}$ .

[F1]

Kernel and image of a linear map: $b \in ran (A)$ means that some $x$ satisfies $b = A x$ .

[F2]

The Moore--Penrose pseudoinverse $A^{+}$ as the solution of the four Penrose equations: $A A^{+} A = A$ ; Every finite real or complex matrix has a unique Moore--Penrose pseudoinverse: this pseudoinverse exists and is unique.

Proof

1.1givenF1F2choosealgebra

By [F1], choose $x$ with $b = A x$ . The Penrose equation in [F2] gives $A A^{+} b = A A^{+} A x = A x = b$ .

2.1givenstep 1.1algebra∎

If $A^{+} b = 0$ , then $A A^{+} b = 0$ , contradicting step 1.1 and the hypothesis $b \neq 0$ . Hence $A^{+} b \neq 0$ and the normalized QLSP target is defined.

Depends on

Used by

HHL finite sine-window phase-estimation algorithm Algorithm

Cited to discharge well-definedness by Quantum linear-systems state-output task.

Dependency tree · two levels

11 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)