The approximate degree of a Boolean function is the minimum degree of real polynomial that approximates it pointwise. For any Boolean function, its approximate degree serves as a lower bound on its quantum query complexity, and generically lifts to a quantum communication lower bound for a related function. We introduce a framework for proving approximate degree lower bounds for certain oracle identification problems, where the goal is to recover a hidden binary string $x \in \{0, 1\}^n$ given possibly non-standard oracle access to it. We apply this framework to the ordered search and hidden string problems, proving nearly tight approximate degree lower bounds of $\Omega(n/\log^2 n)$ for each.

翻译：布尔函数的近似度是逐点逼近该函数的最小实多项式次数。对于任意布尔函数，其近似度构成量子查询复杂度的下界，并通常能提升至关联函数的量子通信下界。我们提出一个用于证明特定预言机识别问题近似度下界的框架，其中目标是在可能通过非标准预言机访问的情况下恢复隐藏二进制字符串 $x \in \{0, 1\}^n$。我们将此框架应用于有序搜索和隐藏字符串问题，为每个问题证明了接近最优的近似度下界 $\Omega(n/\log^2 n)$。