Superlinear lower bounds for triangle detection

Establish a superlinear quantum query lower bound for detecting a triangle in an n-vertex graph, or equivalently obtain a lower bound exceeding the linear scale for the triangle-detection problem; such a result would also imply superlinear lower bounds for every connected non-bipartite pattern.

Background

The paper develops the first unconditional superlinear quantum query lower bounds for detecting fixed subgraphs, but its techniques apply only to cliques of order at least four and certain higher-chromatic patterns. Triangle detection remains outside the reach of these methods. Because every connected non-bipartite pattern is at least as hard as triangle detection under the paper’s reduction, resolving the triangle case would yield consequences for all connected non-bipartite patterns, including odd cycles.

The authors note that the matching-based argument cannot apply directly because a triangle has no matching of size two. They further identify graph collision as a route to progress: an omega(sqrt(n)) lower bound for graph collision on some fixed n-vertex graph would imply a superlinear lower bound for triangle detection.

References

Triangle detection still has no superlinear lower bound. By \Cref{cor:chromatic-comparison}, every connected non-bipartite pattern is at least as hard as the triangle, so a superlinear bound for triangles would give one for every connected non-bipartite pattern, including all odd cycles.

— Superlinear Quantum Query Lower Bounds for Subgraph Detection  (2609.40263 - Gilani et al., 30 Sep 2026) in Section 1, subsection “Open Problems,” paragraph “Triangles and small bipartite patterns”

More generally, which fixed patterns can be detected with $O(n)$ queries? By \Cref{thm:extensions}, no connected pattern of chromatic number at least four can. Among bipartite patterns, both behaviors occur: fixed paths can be detected with $O(n)$ queries, whereas large complete bipartite graphs cannot. For chromatic number three, our results give no superlinear bound, and the triangle is the natural first target.

— Superlinear Quantum Query Lower Bounds for Subgraph Detection  (2609.40263 - Gilani et al., 30 Sep 2026) in Section 1, subsection “Open Problems,” paragraph “Triangles and small bipartite patterns”