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on Does amplitude estimation require the inverse unitary?

The statement

Estimate $|\langle0|U|0\rangle|^2$ to error $\varepsilon$ with constant high success probability for an unknown $d$ -dimensional unitary $U$ . Access to $U$ and $U^\dagger$ allows $O(1/\varepsilon)$ uses via amplitude estimation, whereas independent sampling uses $O(1/\varepsilon^2)$ . Prove a separation between forward-only access and access to both directions, for sufficiently large $d$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.