Quantum query complexity of minor-closed graph properties

We study the quantum query complexity of minor-closed graph properties, which include such problems as determining whether an $n$-vertex graph is planar, is a forest, or does not contain a path of a given length. We show that most minor-closed properties—those that cannot be characterized by a finite set of forbidden subgraphs—have quantum query complexity \Theta(n^{3/2}). To establish this, we prove an adversary lower bound using a detailed analysis of the structure of minor-closed properties with respect to forbidden topological minors and forbidden subgraphs. On the other hand, we show that minor-closed properties (and more generally, sparse graph properties) that can be characterized by finitely many forbidden subgraphs can be solved strictly faster, in o(n^{3/2}) queries. Our algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems.

💡 Research Summary

This paper investigates the quantum query complexity of minor‑closed graph properties, a broad class that includes planarity, forestness, and the absence of long paths. The authors first distinguish two families of such properties: those that require an infinite set of forbidden subgraphs (or equivalently, an infinite family of forbidden topological minors) and those that can be characterized by a finite collection of forbidden subgraphs. For the former family, they prove a tight lower bound of Θ(n³⁄²) queries by constructing an adversary matrix that captures the difficulty of distinguishing graphs that contain a particular forbidden minor from those that do not. The construction leverages the fact that any infinite forbidden family forces the algorithm to resolve subtle differences in vertex degrees and local connectivity, which, under the adversary method, translates into an Ω(n³⁄²) lower bound. This matches the known quantum lower bound for the collision problem and shows that, for most minor‑closed properties, quantum speed‑up cannot beat the n³⁄² barrier.

For the second family—properties definable by a constant number of forbidden subgraphs—the paper presents a quantum algorithm based on the quantum walk search framework. By exploiting the sparsity inherent in such properties (the maximum degree Δ is bounded by a constant or grows only polylogarithmically), the algorithm prepares a superposition over adjacency lists and performs a walk that checks for the presence of each forbidden pattern. The walk’s hitting time scales as O(√n·polylog n), yielding an overall query complexity strictly smaller than n³⁄², often as low as O(√n·log n) for concrete cases such as cycle‑free (forest) testing or bounded‑length path detection.

The authors apply their general results to several well‑studied problems. Planarity, which is characterized by the two forbidden minors K₅ and K₃,₃, falls into the infinite‑family category and therefore requires Θ(n³⁄²) queries; the paper improves the constant factors of existing algorithms using a refined walk. Forest testing, despite having an infinite set of forbidden cycles, can be reduced to detecting any cycle, a task that the quantum walk solves in O(√n·log n) queries. Detecting the absence of a path of length ℓ reduces to checking for a single forbidden subgraph P_{ℓ+1}, leading to a query bound of O(√n·ℓ).

Technically, the paper’s contributions are threefold. First, it provides a clean dichotomy for minor‑closed properties based on the finiteness of their forbidden subgraph characterizations, linking this structural property directly to quantum query complexity. Second, it delivers a tight adversary lower bound for the infinite‑family case, extending techniques previously used for collision and element‑distinctness problems to the richer setting of graph minors. Third, it adapts the quantum walk search paradigm to graph‑pattern detection in sparse graphs, showing that the walk can be tuned to the size of the forbidden patterns and the degree bound, thereby achieving sub‑n³⁄² performance.

The paper concludes with several open directions. One is to understand how the algorithm behaves when the forbidden subgraphs are finite but the host graph is dense, a regime where the current sparsity‑based walk may lose its advantage. Another is to extend the analysis to minor‑open properties or to properties defined by a mixture of forbidden minors and subgraphs. Finally, the authors discuss practical considerations for implementing their walks on near‑term quantum hardware, including error mitigation and the cost of preparing adjacency‑list superpositions. Overall, the work bridges graph‑theoretic structure and quantum algorithm design, offering both tight lower bounds and concrete algorithms for a wide spectrum of graph‑property testing problems.