LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generated

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.

One-query oracle perturbation bound

Statement

For $N \geq 1$ , $1 \leq i \leq N$ , and any vector $ψ$ in the address, response, and work registers, $∥ (O_{e_{i}} - O_{0^{N}}) ψ ∥ \leq 2 ∥ Π_{i - 1} ψ ∥,$ where $e_{i}$ is the input with its only $1$ in coordinate $i$ and $Π_{i - 1}$ projects onto zero-based query address $i - 1$ .

Facts & Assumptions

Given: The standard XOR query from Boolean oracle query model for search lower bounds, the singleton input $e_{i}$ from All-zero versus exactly-one-marked search promise, and the address projector from Query-register projectors and mismatch indicators.

[F1]

Boolean oracle query model for search lower bounds: $O_{x} ∣ j, b, w ⟩ = ∣ j, b \oplus x_{j + 1}, w ⟩$ for $0 \leq j < N$ .

[F2]

Query-register projectors and mismatch indicators: $Π_{i - 1}$ is the orthogonal projection onto address $i - 1$ , tensored with the identity on response and work registers.

[F3]

All-zero versus exactly-one-marked search promise: $e_{i}$ has value $1$ exactly at input coordinate $i$ and $0^{N}$ has value $0$ at every coordinate.

Proof

1.1F1F2F3algebra

On each address block other than $i - 1$ , the two oracle actions in [F1] agree; on block $i - 1$ , the singleton oracle applies Pauli $X$ to the response and the zero oracle applies identity. Hence $O_{e_{i}} - O_{0^{N}} = Π_{i - 1} (X - I) Π_{i - 1}$ , where $X - I$ acts on response and identities act on address and work.

2.1F2step 1.1algebra∎

The orthogonal projection is contractive and $X$ preserves norm because it permutes the two orthonormal response basis vectors; thus $∥ (X - I) Π_{i - 1} ψ ∥ \leq ∥ X Π_{i - 1} ψ ∥ + ∥ Π_{i - 1} ψ ∥ = 2 ∥ Π_{i - 1} ψ ∥$ . Step 1.1 and idempotence of $Π_{i - 1}$ give the claimed bound.

Depends on

Used by

Cumulative hybrid distance for a T-query algorithm Lemma

Dependency tree · two levels

14 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

Charles H. Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani, Strengths and Weaknesses of Quantum Computing (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms (2025) (standard reference, not scraped)