Triangles capture higher-order structures in graphs and are fundamental to applications such as clustering and network analysis. To enable efficient use of such structures at scale, we study the problem of triangle cut sparsification, which aims to reduce the graph size while approximately preserving triangle counts across every cut. We investigate quantum algorithms for this problem, using triangle listing as our main technical ingredient. In particular, we present a quantum algorithm for triangle listing that, for a graph with $n$ vertices, $m$ edges, and $t$ triangles, runs in time $T_{\mathrm{q\text{-}list}} =$ $\widetilde{O}\bigl(\min(n^{5/4}t^{7/12} + n^{7/6}t^{7/9}, m + m^{3/4}t^{1/2},$ $n^{3/2}t^{1/2})\bigr)$, improving upon the best known classical bounds over a broad range of parameters. Our algorithm is based on a heavy-light vertex partition and an extension of triangle detection via quantum walks and Grover search. Leveraging this result, we design a quantum algorithm for constructing $\varepsilon$-triangle cut sparsifiers of size $\widetilde{O}(n/\varepsilon^2)$ in time $\widetilde{O}(T_{\mathrm{q\text{-}list}} + \sqrt{mn}/\varepsilon)$. Finally, we demonstrate applications to clustering algorithms based on triangle-related measures and prove a lower bound of $Ω(n/\varepsilon^2)$ on the size of any $\varepsilon$-triangle cut sparsifiers.

翻译：三角形捕获了图中的高阶结构，是聚类和网络分析等应用的基础。为了实现这类结构在大规模数据上的高效利用，我们研究了三角割稀疏化问题，该问题旨在减少图规模的同时，近似保留每个割上的三角形计数。我们针对这一问题探索了量子算法，并将三角形列出作为主要技术工具。特别地，我们提出了一种用于三角形列出的量子算法，对于具有$n$个顶点、$m$条边和$t$个三角形的图，运行时间为$T_{\mathrm{q\text{-}list}} =$ $\widetilde{O}\bigl(\min(n^{5/4}t^{7/12} + n^{7/6}t^{7/9}, m + m^{3/4}t^{1/2},$ $n^{3/2}t^{1/2})\bigr)$，在广泛参数范围内改进了已知最优经典算法。我们的算法基于重轻顶点划分，并通过量子行走和Grover搜索扩展了三角形检测方法。利用这一结果，我们设计了一种量子算法，用于在时间$\widetilde{O}(T_{\mathrm{q\text{-}list}} + \sqrt{mn}/\varepsilon)$内构造规模为$\widetilde{O}(n/\varepsilon^2)$的$\varepsilon$-三角割稀疏化。最后，我们展示了其在基于三角形度量的聚类算法中的应用，并证了任何$\varepsilon$-三角割稀疏化规模的下界为$Ω(n/\varepsilon^2)$。