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：图稀疏化是大量算法的基础，其应用范围涵盖从分割问题的近似算法到图拉普拉斯线性系统的求解器。在其最强形式中，“谱稀疏化”将边数减少至节点数的近线性水平，同时近似保留图的分割与谱结构。本文展示了谱稀疏化及其众多应用的量子多项式加速。具体而言，我们提出一种量子算法，给定一个包含$n$个节点和$m$条边的加权图，能够在亚线性时间$\tilde{O}(\sqrt{mn}/\epsilon)$内输出一个$\epsilon$-谱稀疏化的经典描述。这一结果与最优经典复杂度$\tilde{O}(m)$形成对比。我们还证明所提出的量子算法在多项式对数因子意义下是最优的。该算法构建于稀疏化、图伸展树、最短路径量子算法及$k$阶独立随机串高效构造的现有研究成果之上。我们的算法意味着在求解拉普拉斯系统以及近似最小割、最稀疏割等一系列分割问题时，可实现量子加速。