Graph sparsification underlies a large number of algorithms, ranging from approximation algorithms for cut problems to solvers for linear systems in the graph Laplacian. In its strongest form, "spectral sparsification" reduces the number of edges to near-linear in the number of nodes, while approximately preserving the cut and spectral structure of the graph. In this work we demonstrate a polynomial quantum speedup for spectral sparsification and many of its applications. In particular, we give a quantum algorithm that, given a weighted graph with $n$ nodes and $m$ edges, outputs a classical description of an $\epsilon$-spectral sparsifier in sublinear time $\tilde{O}(\sqrt{mn}/\epsilon)$. This contrasts with the optimal classical complexity $\tilde{O}(m)$. We also prove that our quantum algorithm is optimal up to polylog-factors. The algorithm builds on a string of existing results on sparsification, graph spanners, quantum algorithms for shortest paths, and efficient constructions for $k$-wise independent random strings. Our algorithm implies a quantum speedup for solving Laplacian systems and for approximating a range of cut problems such as min cut and sparsest cut.
翻译:图形垃圾化是大量算法的基础, 从截断问题的近似算法到图 Laplacian 中的线性系统的解析器。 最强烈的形式是, “ 光谱封闭化” 将节点数的边缘数减到近线性, 并大致保留图的剪切和光谱结构 。 在这项工作中, 我们展示了一个用于光谱封闭及其许多应用的多元量子加速度。 特别是, 我们给出了量子算法, 根据一个带有美元节点和美元边缘的加权图形, 算法输出一个在亚线性时间 $\ tilde{ O} (\\\ qrt{mn}/\ epsilon) 中 的美元光谱质谱化器的典型描述 。 这与最理想的经典复杂性 $\ tilde{ O} (m) 和许多应用的量子量算法是最佳的。 我们的量子算算法是以 最优的 。 。 该算法建立在一个在蒸气化、 平面仪仪仪仪仪、 度算算算算算算算法 和 直径解的精度解的系统 解的精度解速度问题。