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.

翻译：摘要：函数评估的标准对手界指出，在量子查询模型中，要以成功概率 $\frac{1}{2}+\delta$ 评估函数 $f\colon D\to C$，至少需要 $\left( 2\delta -\sqrt{1-4\delta^2} \right) Adv(f)$ 次查询，其中 $Adv(f)$ 是某个优化问题的最优值。对于 $\delta \ll 1$，即使经过重复并采用多数投票论证，该界限也仅为 $\Theta\left(\delta^2 Adv(f)\right)$。相比之下，多项式方法有时能证明一个在 $\delta \to 0$ 时不收敛于 $0$ 的界限。我们改进了依赖于 $\delta$ 的前置因子，并实现了 $2\delta Adv(f)$ 的界限。证明思路是“将输出条件转化为输入条件”：从一个能将完全输入无关的初始态完美转化为不完全可区分的最终态的算法出发，我们通过投影到“正确”的最终子空间并消去计算，构造出在相同查询次数内将不完全输入无关的初始态转化为完全可区分的最终态的算法。由此得到的依赖于 $\delta$ 的初始 Gram 矩阵条件，与原始算法中最终 Gram 矩阵的条件相比，能够推导出收紧的前置因子。