Close the quantum triangle-listing query-complexity gap

Close the gap between the quantum upper bound \(\widetilde{O}(n^{35/18})\) and the lower bound \(\Omega(n^{3/2-o(1)})\) for listing a constant fraction of the \(\Theta(n)\) triangles in dense \(n\)-vertex graphs in the mixed quantum graph-query model.

Background

The paper proves the first nontrivial quantum query lower bound for triangle listing, namely Ω(n3/2−o(1))\Omega(n^{3/2-o(1)}) on graphs with Θ(n)\Theta(n) triangles. For graphs of the relevant density, the best known quantum algorithm uses O~(n35/18)\widetilde{O}(n^{35/18}) queries, leaving a substantial gap between the known upper and lower bounds.

References

On graphs of the relevant density ($n{2-o(1)}$ edges with $t=\Theta(n)$ triangles), the best known upper bound is $\tilde{O}(n{5/4}t{7/12}+n{7/6} t{7/9})=\tilde{O}(n{35/18})$ ; closing the gap between $\tilde{O}(n{35/18})$ and $\Omega(n{3/2-o(1)})$ remains an interesting open problem.

— Quantum Query Lower Bounds for Triangle-Listing and Spanners  (2609.37091 - Chen et al., 29 Sep 2026) in Section 1, paragraph “Triangle Listing”