Tarski's theorem states that every monotone function from a complete lattice to itself has a fixed point. We specifically consider the two-dimensional lattice $\mathcal{L}^2_n$ on points $\{1, \ldots, n\}^2$ and where $(x_1, y_1) \leq (x_2, y_2)$ if $x_1 \leq x_2$ and $y_1 \leq y_2$. We show that the quantum query complexity of finding a fixed point given query access to a monotone function on $\mathcal{L}^2_n$ is $Ω((\log n)^2)$, matching the classical deterministic upper bound. The proof consists of two main parts: a lower bound on the quantum query complexity of a composition of a class of functions including ordered search, and an extremely close relationship between finding Tarski fixed points and nested ordered search.

翻译：塔尔斯基定理指出，每个从完备格到自身的单调函数都有不动点。我们特别考虑二维格 $\mathcal{L}^2_n$，其顶点集为 $\{1, \ldots, n\}^2$，且当 $x_1 \leq x_2$ 且 $y_1 \leq y_2$ 时，有 $(x_1, y_1) \leq (x_2, y_2)$。我们证明，在具有查询访问$\mathcal{L}^2_n$上单调函数的条件下，寻找不动点的量子查询复杂度为 $Ω((\log n)^2)$，这与经典确定性上界相匹配。证明主要由两部分组成：对一类包含有序搜索的函数组合的量子查询复杂度下界，以及寻找塔尔斯基不动点与嵌套有序搜索之间的极其紧密的联系。