Quantum algorithms for path and cycle containment problems

2026-05-09

AI Summary

This work investigates the quantum query complexity of subgraph containment problems for constant-length paths and cycles in graphs. Considering both directed and undirected graphs represented via adjacency matrices, and accounting for variations such as exact versus at-most length constraints and promise conditions, the study employs randomized reductions to elucidate relative problem hardness and establishes a dichotomy for path containment, identifying a class of nonlinearly equivalent problems. Leveraging conditional lower bounds derived from quantum walk techniques and the graph collision problem, the authors design a novel quantum algorithm that improves the query complexity from the previously known $O(n^{3/2})$ to $\tilde{O}(n^{3/2 - \alpha_k})$, where $\alpha_k \in \Theta(c^{-k})$ with $c \approx 1.33$. They further prove that no linear-query quantum algorithm can efficiently solve these problems unless the graph collision problem admits an $O(\sqrt{n})$-query algorithm.

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 $\widetilde{O}(n^{3/2-α_k})$, 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.

Problem

• quantum query complexity
• subgraph containment
• path detection
• cycle detection
• adjacency matrix model

Innovation

• quantum walk
• query complexity
• subgraph containment
• path detection
• cycle detection