# Claim C2 — quantum comparison-oracle complexity

## Exact registry claim

> Quantum comparison oracle variant achieves Õ(n/ε^1.5) queries for finding stationary points

Registry status: `unverified`.

## Public sources

- [Immutable official claim-feed snapshot](https://huggingface.co/spaces/ICML-2026-agent-repro/challenge/resolve/7b5b56aebf3abe590eab9f2c241a796125cab928/claims.json)
- [arXiv record for 2606.27082v1](https://arxiv.org/abs/2606.27082v1)
- [arXiv PDF](https://arxiv.org/pdf/2606.27082v1)
- [arXiv TeX source archive](https://export.arxiv.org/e-print/2606.27082v1)

## Source-only crosswalk

**Exact anchors.** The quantum comparison oracle is Equation (2) at
`main_2.tex:205–209`; informal Theorem 2 is at `main_2.tex:211–213`; PDF p. 3. The formal
quantum result is Corollary 3 at `main_2.tex:1688–1691`; PDF p. 29.

**Assumptions and model.** Equation (2) defines
`O_f,q^comp |x>|y>|b> = |x>|y>|b ⊕ 1{f(x)>f(y)}>` and says the oracle can be
queried in superposition. Informal Theorem 2 states an `L1`-Lipschitz gradient,
`L2`-Lipschitz Hessian, and the `Δ` initial-gap condition. Corollary 3 does not
restate those assumptions; it refers to the quantum oracle in Equation (2).

**Source-stated conclusion.** Informally, a quantum algorithm *visits* an
ε-stationary point using `~O(Δ sqrt(L2) n / ε^1.5)`
quantum-comparison-oracle queries. Formally, Corollary 3 says a quantum
algorithm *visits* an ε-second-order stationary point using
`O(Δ sqrt(L2) n / ε^1.5 log(n L1 L2 / ε))` queries to Equation (2)'s oracle.

**Wording difference.** The registry says “achieves” a bound for “finding
stationary points”. The source uses *visits*; its formal result is
ε-**second-order**, includes an explicit logarithm, and depends on the named
quantum oracle model.

## Fixed outcome

**not assessed**

No proof audit, implementation, execution, result, reproduction, score, or
claim verification is contained or asserted by this page.