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.

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.

Odd-well and even-ill reciprocal filter

Definition

For $κ \geq 1$ and real estimate $z$ , put $u = ∣ z ∣$ and define the odd well amplitude $f_{κ}$ and even ill amplitude $g_{κ}$ by $f_{κ} (z) = {\begin{cases} sgn (z) / (2 κ u), & u \geq 1 / κ, \\ \frac{1}{2} sgn (z) \sin (π κ (u - 1 / (2 κ))), & 1 / (2 κ) \leq u < 1 / κ, \\ 0, & u < 1 / (2 κ), \end{cases} g_{κ} (z) = {\begin{cases} 0, & u \geq 1 / κ, \\ \frac{1}{2} \cos (π κ (u - 1 / (2 κ))), & 1 / (2 κ) \leq u < 1 / κ, \\ 1 / 2, & u < 1 / (2 κ), \end{cases}$ where $sgn (z) = 1$ for $z > 0$ , $- 1$ for $z < 0$ , and $0$ for $z = 0$ . The three flag amplitudes are $(h (z), f_{κ} (z), g_{κ} (z))$ , where $h (z) = \sqrt{1 - f_{κ} (z)^{2} - g_{κ} (z)^{2}}$ . Labels are nothing, well, ill, respectively. This definition applies to every real decoded grid point, including points outside the supported spectrum.

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

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)