We consider the quantum query complexity of local search as a function of graph geometry. Given a graph $G = (V,E)$ with $n$ vertices and black box access to a function $f : V \to \mathbb{R}$, the goal is find a vertex $v$ that is a local minimum, i.e. with $f(v) \leq f(u)$ for all $(u,v) \in E$, using as few oracle queries as possible. We show that the quantum query complexity of local search on $G$ is $\Omega\bigl( \frac{n^{\frac{3}{4}}}{\sqrt{g}} \bigr)$, where $g$ is the vertex congestion of the graph. For a $\beta$-expander with maximum degree $\Delta$, this implies a lower bound of $ \Omega\bigl(\frac{\sqrt{\beta} \; n^{\frac{1}{4}}}{\sqrt{\Delta} \; \log{n}} \bigr)$. We obtain these bounds by applying the strong weighted adversary method to a construction by Br\^anzei, Choo, and Recker (2024). As a corollary, on constant degree expanders, we derive a lower bound of $\Omega\bigl(\frac{n^{\frac{1}{4}}}{ \sqrt{\log{n}}} \bigr)$. This improves upon the best prior quantum lower bound of $\Omega\bigl( \frac{n^{\frac{1}{8}}}{\log{n}}\bigr) $ by Santha and Szegedy (2004). In contrast to the classical setting, a gap remains in the quantum case between our lower bound and the best-known upper bound of $O\bigl( n^{\frac{1}{3}} \bigr)$ for such graphs.

翻译：我们考虑局部搜索的量子查询复杂度作为图几何结构的函数。给定一个图 $G = (V,E)$，其具有 $n$ 个顶点，以及对函数 $f : V \to \mathbb{R}$ 的黑盒访问，目标是以尽可能少的预言查询次数找到一个局部最小顶点 $v$，即对于所有 $(u,v) \in E$ 满足 $f(v) \leq f(u)$。我们证明，在图 $G$ 上进行局部搜索的量子查询复杂度为 $\Omega\bigl( \frac{n^{\frac{3}{4}}}{\sqrt{g}} \bigr)$，其中 $g$ 是图的顶点拥塞度。对于一个最大度为 $\Delta$ 的 $\beta$-扩展器，这蕴含了一个 $ \Omega\bigl(\frac{\sqrt{\beta} \; n^{\frac{1}{4}}}{\sqrt{\Delta} \; \log{n}} \bigr)$ 的下界。我们通过将强加权对手方法应用于 Br\^anzei、Choo 和 Recker（2024）的一个构造来得到这些下界。作为一个推论，在常数度扩展器上，我们推导出一个 $\Omega\bigl(\frac{n^{\frac{1}{4}}}{ \sqrt{\log{n}}} \bigr)$ 的下界。这改进了 Santha 和 Szegedy（2004）先前最好的量子下界 $\Omega\bigl( \frac{n^{\frac{1}{8}}}{\log{n}}\bigr)$。与经典情形相反，在量子情形下，我们的下界与对此类图已知的最佳上界 $O\bigl( n^{\frac{1}{3}} \bigr)$ 之间仍然存在差距。