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-Z and SWAP

Definition

In the ordered two-qubit basis $∣ 00 ⟩, ∣ 01 ⟩, ∣ 10 ⟩, ∣ 11 ⟩$ , define

$C Z : = C P (π) = diag (1, 1, 1, - 1) .$

Define the SWAP operator on the ordered tensor product by $SWAP (∣ a ⟩ \otimes ∣ b ⟩) = ∣ b ⟩ \otimes ∣ a ⟩$ for $a, b \in {0, 1}$ , and extend this rule linearly from the ordered product basis. SWAP is an operator that permutes the register state; it does not relabel the ordered wires.

Facts & Assumptions

Given: The ordered two-qubit computational basis and the definitions of $C P (θ)$ and the tensor product register.

[L1]

Controlled Phase Rotation defines $C P (θ) = diag (1, 1, 1, e^{i θ})$ in control-first order. Its carrier is present in this batch.

[L2]

Ordered tensor registers, their inner product, and wire permutations defines the ordered factors and product basis. Its current draft proof was read.

[L3]

The computational basis is orthonormal proves the computational basis is orthonormal. Its current draft proof was read, including the one-dimensional zero-qubit boundary case.

[F1]

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

Verification

1.1givenL1F1algebra

Substituting $θ = π$ into [L1] gives $C P (π) = diag (1, 1, 1, e^{i π}) = diag (1, 1, 1, - 1)$ by [F1].

1.2givenL2L3algebra

SWAP fixes $∣ 00 ⟩$ and $∣ 11 ⟩$ and interchanges $∣ 01 ⟩$ with $∣ 10 ⟩$ . The ordered product basis in [L2] spans the tensor product, so these actions determine a unique linear map. By [L2] and [L3] it permutes an orthonormal basis, hence preserves norms and is its own inverse; therefore it is unitary.

2.1step 1.2∎

The preceding action changes the state vector in the fixed ordered tensor product, while the first and second factors remain the declared wires. Thus the definition is an actual linear gate action, not a change of names for the tensor factors.

Depends on

Used by

Exact success amplification by attenuating the good branch Algorithm
Stabilizer encoder by symplectic elimination Algorithm

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

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)