TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck pass

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.

Phase kickback from the standard bit query

Statement

Let $f : {0, 1}^{a} \to {0, 1}$ be a total Boolean function. If the one-bit target of its standard query is prepared in

∣ - ⟩ : = \frac{∣ 0 ⟩ - ∣ 1 ⟩}{\sqrt{2}},

then one standard query acts on every address basis vector by

U_{f} (∣ x ⟩ ∣ - ⟩) = (- 1)^{f (x)} ∣ x ⟩ ∣ - ⟩ .

Thus the address acquires exactly the phase query $P_{f}$ , while the target returns unchanged. The same identity holds for every address superposition.

Facts & Assumptions

Given: The total Boolean function, the standard XOR query, and the normalized target vector $∣ - ⟩$ .

[F1]

Standard bit-flip query convention: $U_{f} ∣ x ⟩ ∣ y ⟩ = ∣ x ⟩ ∣ y \oplus f (x) ⟩$ .

[F2]

Binary phase-query convention: $P_{f} ∣ x ⟩ = (- 1)^{f (x)} ∣ x ⟩$ , extended linearly.

[F3]

Pure state and ray: a normalized nonzero vector represents a pure state; the displayed unit vector is an allowed target state.

[F4]

Finite registers and their ordered computational bases, The computational basis is orthonormal and The product basis is orthonormal and tensor dimensions multiply: the ordered computational-basis vectors span each finite address register, including the zero-qubit register.

[F5]

Dirac Notation Conversion: ket notation denotes the corresponding vectors with the repository's stated inner-product convention.

Proof

1.1givenF1F3algebra

If $f (x) = 0$ , the query leaves the target bits unchanged, so it fixes $∣ x ⟩ ∣ - ⟩$ . If $f (x) = 1$ , it swaps the target basis vectors and sends $(∣ 0 ⟩ - ∣ 1 ⟩) / \sqrt{2}$ to $(∣ 1 ⟩ - ∣ 0 ⟩) / \sqrt{2} = - ∣ - ⟩$ . Therefore in both cases the target returns unchanged and the scalar is $(- 1)^{f (x)}$ .

2.1step 1.1F2F4F5algebra∎

By [F2], the address factor in step 1.1 is exactly $P_{f} ∣ x ⟩$ . Any address vector is a finite linear combination $\sum_{x} α_{x} ∣ x ⟩$ by [F4]; linearity of $U_{f}$ gives $\sum_{x} α_{x} (- 1)^{f (x)} ∣ x ⟩ ∣ - ⟩ = (P_{f} \sum_{x} α_{x} ∣ x ⟩) \otimes ∣ - ⟩$ . Hence the target is unchanged and the identity holds on every superposition.

Depends on

Used by

Oracle conversion and uncomputation examples Example
One standard query implements one phase query Theorem

Dependency tree · two levels

27 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 Information, Chapter 6: Quantum Algorithms (updated November 2020) (standard reference, not scraped)