ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-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.

Unique-collision element-distinctness parameters

Example

Suppose exactly one pair of distinct indices has equal values. Compute its marked fraction for $N = 8, r = 4$ , and verify the balanced setup/update expression for $r$ of order $N^{2 / 3}$ .

Facts & Assumptions

Given: The Johnson-walk state and cost conventions of Element-distinctness Johnson walk and value access and the query bound of Element-distinctness O(N^{2/3}) query bound.

[F1]

Element-distinctness Johnson walk and value access: a state is an $r$ -subset, and its marked condition is the presence of a colliding pair; setup and update query costs are $r$ and $2$ .

[F2]

Element-distinctness O(N^{2/3}) query bound: a fixed pair is included with probability $r (r - 1) / [N (N - 1)]$ , and the one-outer-trial query bound is $2 r + O (N / \sqrt{r})$ .

Proof

1.1F1F2algebra

For $N = 8, r = 4$ , the state space has $(\binom{8}{4}) = 70$ subsets. Exactly $(\binom{6}{2}) = 15$ contain the unique colliding pair, so $ε = 15 / 70 = 3 / 14$ , equal to $r (r - 1) / [N (N - 1)] = 12 / 56$ . The stationary mass is positive, and [F1] gives four setup value queries, two value queries per update, and no extra value query for the stored-value check.

2.1F1F2step 1.1algebra∎

More generally, for $r = Θ (N^{2 / 3})$ with $r \leq N / 2$ , [F2] charges at most two fresh setups, with leading term $2 r = Θ (N^{2 / 3})$ and update/search term $N / \sqrt{r} = Θ (N^{2 / 3})$ . Thus both terms are of the same asymptotic order and the theorem's total is $O (N^{2 / 3})$ . For $N = 8$ , the clamp in the theorem selects $r = 4$ ; its expression is $8 + O (8 / 2) = 8 + O (4)$ , consistent with this scaling.

Depends on

Used by

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Dependency tree · two levels

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Sources

Andris Ambainis, Quantum walk algorithm for element distinctness, Theorems 4–5 and §§3–4 (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms, Chapter 20, §§20.1–20.3 (standard reference, not scraped)