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.

Aggregate hybrid distance budget over all marked indices

Statement

For every $T \geq 0$ query algorithm on $N$ addresses, with final states and query magnitudes as in Cumulative hybrid distance for a T-query algorithm, $\sum_{i = 1}^{N} ∥ ψ_{T}^{i} - ψ_{T}^{0} ∥^{2} \leq 4 T^{2} .$

Facts & Assumptions

Given: The cumulative hybrid estimate and the complete orthogonal address projectors.

[F1]

Cumulative hybrid distance for a T-query algorithm: $∥ ψ_{T}^{i} - ψ_{T}^{0} ∥ \leq 2 \sum_{t < T} \sqrt{q_{t, i}}$ with $q_{t, i} = ∥ Π_{i - 1} ψ_{t}^{0} ∥^{2}$ .

[F2]

Query-register projectors and mismatch indicators: $Π_{0}, \dots, Π_{N - 1}$ are pairwise orthogonal and sum to identity on the query address, response, and work registers.

[F3]

Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs: for finite real vectors, the square of the sum of coordinates is at most the vector length times the sum of their squares.

Proof

1.1F1F3algebra

If $T = 0$ , [F1] gives zero distance for every $i$ , so the sum is zero. If $T \geq 1$ , squaring [F1] and applying [F3] to the real numbers $\sqrt{q_{t, i}}$ gives $∥ ψ_{T}^{i} - ψ_{T}^{0} ∥^{2} \leq 4 T \sum_{t = 0}^{T - 1} q_{t, i}$ ; each $q_{t, i}$ is nonnegative because it is a squared norm.

2.1F2step 1.1algebra∎

Summing over $i$ and interchanging finite sums, completeness and orthogonality in [F2] give $\sum_{i = 1}^{N} q_{t, i} = \sum_{j = 0}^{N - 1} ∥ Π_{j} ψ_{t}^{0} ∥^{2} = ∥ ψ_{t}^{0} ∥^{2} = 1$ for each $t$ . Therefore $\sum_{i} ∥ ψ_{T}^{i} - ψ_{T}^{0} ∥^{2} \leq 4 T \sum_{t < T} 1 = 4 T^{2}$ .

Depends on

Used by

BBBV hybrid lower bound for unstructured search Theorem

Dependency tree · two levels

16 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)