ExampleConstruction: AI-adaptedVerification: 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.

Two-dimensional Hermitian block-encoding signal

Example

Fix $α > 0$ , $x \in [- 1, 1]$ , $s = \sqrt{1 - x^{2}}$ , and take one-dimensional system $H = C$ with $K = [α x]$ . On a one-qubit ancilla define $U_{x} = (\begin{array}{cc} x & s \\ s & - x \end{array}), J z = ∣ 0 ⟩ z, Π = ∣ 0 ⟩ ⟨ 0 ∣ .$ This is an exact Hermitian block encoding of $K / α = [x]$ . Its signal complement has amplitude $s$ ; at $x = 0$ the signal matrix is $X$ , while at $x = \pm 1$ the clean line is invariant and the normalized complement is not defined or needed.

Facts & Assumptions

Given: $α > 0$ , $x \in [- 1, 1]$ , and the standard orthonormal ancilla basis $∣ 0 ⟩, ∣ 1 ⟩$ .

[F1]

Hermitian block-encoding signal subspaces: for a normalized Hermitian block eigenvalue $x$ , the non-endpoint signal matrix has entries $x, \sqrt{1 - x^{2}}$ , and the endpoint complement is absent without division by zero.

Proof

Given: As in Facts & Assumptions.

1.1givenalgebra

Since $s^{2} = 1 - x^{2}$ , direct multiplication gives $U_{x}^{*} = U_{x}$ and $U_{x}^{2} = I$ . Also $J^{*} U_{x} J = [x] = K / α$ , so $U_{x}$ is unitary and has the claimed block encoding. Its clean and complementary components are $U_{x} ∣ 0 ⟩ = x ∣ 0 ⟩ + s ∣ 1 ⟩$ .

2.1F1step 1.1algebra

If $∣ x ∣ < 1$ , then $s > 0$ , so the normalized complement defined in [F1] is $∣ 1 ⟩$ ; both input and output paired bases are $(∣ 0 ⟩, ∣ 1 ⟩)$ , and the matrix is exactly $U_{x}$ . At $x = 0$ , $s = 1$ , hence $U_{0} = X$ , confirming that zero is a regular two-dimensional signal plane.

3.1F1step 1.1algebra∎

If $x = 1$ or $x = - 1$ , then $s = 0$ and $U_{x} ∣ 0 ⟩ = x ∣ 0 ⟩ = U_{x}^{*} ∣ 0 ⟩$ . No complement vector is normalized in these cases; this verifies the endpoint convention and the encoded eigenvalue $λ / α = x$ .

Depends on

Hermitian block-encoding signal subspaces

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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, Chapter 29 §29.1 (standard reference, not scraped)