Quantum algorithms for path and cycle containment problems

Abstract: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 $\widetilde{O}(n^{3/2-\alpha_k})$, where $\alpha_k \in \Theta(c^{-k})$ 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.

Quantum Physics (quant-ph); Computational Complexity (cs.CC)

arXiv:2605.09017 [quant-ph]

(or arXiv:2605.09017v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2605.09017 Focus to learn more arXiv-issued DOI via DataCite

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