ExampleConstruction: AI-generatedVerification: 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.

A Two-Level Rotation on Two Qubits

Example

Use the ordered computational basis $∣ 00 ⟩, ∣ 01 ⟩, ∣ 10 ⟩, ∣ 11 ⟩$ . Select $x = 00$ , $y = 01$ , and $V = \frac{1}{\sqrt{2}} (\begin{array}{cc} 1 & - 1 \\ 1 & 1 \end{array}) .$ This is the QC7 rotation $R_{y} (π / 2)$ , with its exact sign convention recorded in Exact one-qubit gates and rotations.

Facts & Assumptions

Given: The ordered basis above and the two-level definition.

[F1]

Two-Level Unitary: the matrix $V$ acts on the ordered pair $(∣ 00 ⟩, ∣ 01 ⟩)$ and the operator is identity on the orthogonal complement.

[F2]

One-Qubit Euler Decomposition: $R_{y} (t) = (\begin{matrix} \cos (t / 2) & - \sin (t / 2) \\ \sin (t / 2) & \cos (t / 2) \end{matrix})$ .

Verification

1.1givenF2algebra

Substituting $t = π / 2$ into [F2] gives the displayed $V$ . Its columns each have squared norm $(1 + 1) / 2 = 1$ and inner product $(- 1 + 1) / 2 = 0$ . Hence $V^{*} V = I_{2}$ , so the selected block is unitary.

2.1F1step 1.1algebra∎

Inserting this block at basis positions $∣ 00 ⟩, ∣ 01 ⟩$ and fixing $∣ 10 ⟩, ∣ 11 ⟩$ gives $T_{00, 01} (V) = (\begin{array}{cccc} 1 / \sqrt{2} & - 1 / \sqrt{2} & 0 & 0 \\ 1 / \sqrt{2} & 1 / \sqrt{2} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}) .$ Its exact entries show the identity action on the orthogonal complement and retain no hidden phase quotient. The matrix is exact, not merely equal up to scalar phase.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

26 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

Adriano Barenco et al., Elementary gates for quantum computation, arXiv:quant-ph/9503016v1 (standard reference, not scraped)