DefinitionDefinition: AI-adaptedProof: 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.

Controlled Phase Rotation

Definition

For a real angle $θ$ , define the one-qubit phase matrix $P_{θ} = diag (1, e^{i θ})$ . In control-first order define

$C P (θ) : = C (P_{θ}) = diag (1, 1, 1, e^{i θ})$

in the ordered basis $∣ 00 ⟩, ∣ 01 ⟩, ∣ 10 ⟩, ∣ 11 ⟩$ . This is the exact controlled phase; no circuit cost or controlled access to an arbitrary black-box unitary is implied.

Facts & Assumptions

Given: A real $θ$ , an ordered control-target qubit pair, and the exact phase convention of the Definition.

[L1]

Controlled Unitary Access defines the control-first block action $C (U) = diag (I, U)$ . Its carrier is present in this batch.

[L2]

Exact one-qubit gates and rotations defines the exact $R_{z} (θ)$ exponential and computational basis. Its current draft definition body was read; its item is now present in the QC07 manifest and page; the page remains draft.

[L3]

Pauli formula for an axis rotation proves $R_{z} (θ) = diag (e^{- i θ / 2}, e^{i θ / 2})$ by the convergent Pauli exponential series. Its current draft proof was read; its item is now present in the QC07 manifest and page; the page remains draft.

[L4]

The computational basis is orthonormal proves the ordered computational basis of $Q_{2}$ is orthonormal; this supplier is in the dependency closure through [L3].

[F1]

$\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ : for real $θ$ , $∣ e^{i θ} ∣ = 1$ and $e^{i π} = - 1$ .

[F2]

$\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential: $\exp (z + w) = \exp z \exp w$ for complex $z, w$ .

Verification

1.1givenL1L2L4F1algebra

The target computational basis is orthonormal by [L4], and [F1] gives modulus one for both diagonal entries of $P_{θ}$ . Thus $P_{θ}$ is an invertible isometry and hence unitary, so [L1] applies to its controlled block. The first control block applies the identity, so $∣ 00 ⟩$ and $∣ 01 ⟩$ are fixed. In the second block, $P_{θ}$ fixes $∣ 10 ⟩$ and sends $∣ 11 ⟩$ to $e^{i θ} ∣ 11 ⟩$ . These four actions give the diagonal matrix in the Definition.

2.1step 1.1F1algebra

Every diagonal entry of $C P (θ)$ has modulus one by [F1]; its conjugate transpose is therefore its inverse, so it is unitary.

3.1L1L3F2algebra∎

By [L3], $R_{z} (θ) = diag (e^{- i θ / 2}, e^{i θ / 2})$ . By [F2], multiplying it by $e^{i θ / 2}$ gives $P_{θ} = diag (1, e^{i θ})$ . Therefore $C P (θ) = diag (I, e^{i θ / 2} R_{z} (θ))$ , which is the block phase $diag (I, e^{i θ / 2} I)$ applied after $C (R_{z} (θ))$ . Dropping that control phase would change the defined operation.

Depends on

Used by

Controlled-Z and SWAP Definition
Finite open-graph adaptive measurement pattern Definition
Graph state of an ordered finite simple graph Definition
Normalized transport schedule for a J/CZ circuit Definition
QAOA circuit and expected combinatorial objective Definition
QFT ideal and finite-set gate-cost models Definition
Binary factorization of QFT basis output Lemma
Computational-basis amplitudes of a graph state Lemma
Controlled-Z edge gates commute Lemma
Controlled-Z matching action on a Pauli frame Lemma
Four-branch open-graph pattern for CZ Lemma
Graph-state stabilizer generators Lemma
One-bit J-gate teleportation branch identity Lemma
Circuit and pattern translations are branch preserving and universal Theorem
Controlled-Phase Approximation and Global-Phase Boundary Theorem
Finite universal-gate compilation cost for the truncated QFT Theorem
Graph state is the unique joint stabilizer ray Theorem
Pattern branch probabilities and zero-probability transcripts Theorem

Dependency tree · two levels

34 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

Ronald de Wolf, Quantum Computing: Lecture Notes, arXiv:1907.09415v6 (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 6: Quantum Algorithms (standard reference, not scraped)