The quantum query complexity of subgraph-containment problems, which ask whether a given subgraph $H$ is present in an input graph $G$, has been the subject of considerable study. However, even for relatively simple subgraphs, such as paths and cycles, a complete understanding of their query complexities remains elusive. In this work, we consider several variants of path- and cycle-containment problems in the adjacency matrix model, where we search for paths or cycles of constant length $k$. We compare the settings where the graphs are directed or undirected, where the goal is to detect or find the existence of a path/cycle, and where the path/cycle we are looking for has length exactly $k$, or at most $k$. We also consider several promise versions of these problems, where we suppose that the input graph has a certain structure. We characterize the relative difficulty of these variants of the path/cycle-containment problems, by relating them to one another using randomized reductions, and grouping them into equivalence classes. When we restrict our attention to path-containment problems, we get a dichotomy result. Some of the path-containment problems can be solved using a linear number of queries, and all the others are equivalent to one another (and additionally to several cycle-containment problems) under randomized reductions, up to constant overhead. For the latter equivalence class, we prove a novel quantum-walk-based algorithm that achieves query complexity $\widetilde{O}(n^{3/2-α_k})$, where $α_k \in Θ(c^{-k})$ and $c = \sqrt{3+\sqrt{17}}/2 \approx 1.33$, beating the previous best upper bound $O(n^{3/2})$ on its query complexity. We also provide a conditional lower bound based on the graph-collision problem, which implies that this equivalence class does not admit linear-query quantum algorithms unless graph collision admits an $O(\sqrt{n})$ query algorithm.

翻译：子图包含问题询问给定子图$H$是否存在于输入图$G$中，其量子查询复杂度一直是研究焦点。然而，即使对于路径和环等相对简单的子图，其查询复杂性的完整认知仍悬而未决。本文在邻接矩阵模型下研究了路径与环包含问题的若干变体，聚焦于固定长度$k$的路径或环搜索。我们比较了有向图与无向图场景、检测与存在性判定目标、以及待搜索路径/环长度严格为$k$或不超过$k$等不同设置。同时考虑了输入图具有特定结构的若干承诺版本问题。通过随机化归约建立变体间的关联并将其划分为等价类，我们刻画了路径/环包含问题各变体的相对难度。当限定于路径包含问题时，得到二分性结论：部分路径包含问题可通过线性次查询求解，其余所有问题在常数开销下通过随机化归约相互等价（且等价于若干环包含问题）。针对后者等价类，我们提出基于量子行走的新型算法，实现查询复杂度$\widetilde{O}(n^{3/2-α_k})$，其中$α_k \in Θ(c^{-k})$且$c = \sqrt{3+\sqrt{17}}/2 \approx 1.33$，突破了此前最优上界$O(n^{3/2})$。基于图碰撞问题的条件性下界表明，该等价类不存在线性查询量子算法，除非图碰撞问题存在$O(\sqrt{n})$查询算法。