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. Our lower bounds apply to decision versions of these problems, where the goal is to compute the parity of $x$. We apply our framework to the ordered search and hidden string problems, proving nearly tight approximate degree lower bounds of $\Omega(n/\log^2 n)$ for each. These lower bounds generalize to the weakly unbounded error setting, giving a new quantum query lower bound for the hidden string problem in this regime. Our lower bounds are driven by randomized communication upper bounds for the greater-than and equality functions.

翻译：布尔函数的近似度是逐点逼近该函数的实多项式的最小次数。对于任何布尔函数，其近似度为其量子查询复杂度提供下界，并通常可提升为相关函数的量子通信下界。我们提出一个框架，用于证明特定预言识别问题的近似度下界，这些问题的目标是在可能非标准预言访问下恢复隐藏的二进制字符串$x \in \{0, 1\}^n$。我们的下界适用于这些问题的决策版本，即计算$x$的奇偶性。我们将该框架应用于有序搜索和隐藏字符串问题，分别为每个问题证明近似度下界接近$\Omega(n/\log^2 n)$。这些下界可推广至弱无界错误设置，在该情况下为隐藏字符串问题提供了新的量子查询下界。我们的下界受大于函数和相等函数的随机通信上界驱动。