One Weird Trick Tightens the Quantum Adversary Bound, Especially for Success Probability Close to $1/2$

The textbook adversary bound for function evaluation states that to evaluate a function $f\colon D\to C$ with success probability $\frac{1}{2}+\delta$ in the quantum query model, one needs at least $\left( 2\delta -\sqrt{1-4\delta^2} \right) Adv(f)$ queries, where $Adv(f)$ is the optimal value of a certain optimization problem. For $\delta \ll 1$, this only allows for a bound of $\Theta\left(\delta^2 Adv(f)\right)$ even after a repetition-and-majority-voting argument. In contrast, the polynomial method can sometimes prove a bound that doesn't converge to $0$ as $\delta \to 0$. We improve the $\delta$-dependent prefactor and achieve a bound of $2\delta Adv(f)$. The proof idea is to "turn the output condition into an input condition": From an algorithm that transforms perfectly input-independent initial to imperfectly distinguishable final states, we construct one that transforms imperfectly input-independent initial to perfectly distinguishable final states in the same number of queries by projecting onto the "correct" final subspaces and uncomputing. The resulting $\delta$-dependent condition on initial Gram matrices, compared to the original algorithm's condition on final Gram matrices, allows deriving the tightened prefactor.