QIP2021 | Quantum speedups for graph sparsification, graph cut problems... (Simon Apers)

Quantum speedups for graph sparsification, graph cut problems and Laplacian solving

Authors: Simon Apers, Troy Lee and Ronald de Wolf
Affiliations: ULB - CWI | University of Technology, Sydney | QuSoft, CWI and University of Amsterdam (Amsterdam)

Abstract
Graph sparsification underlies a large number of algorithms for graph cut problems and solving Laplacian systems. 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. We give a quantum algorithm that outputs a classical description of an $\varepsilon$-spectral sparsifier of a weighted graph with $n$ vertices and $m$ edges. It has time complexity $\widetilde O(\sqrt{mn}/\varepsilon)$ in the adjacency array model and $\widetilde O(n^{3/2}/\varepsilon)$ in the adjacency matrix model. These bounds are tight up to polylogarithmic factors, and improve on the optimal classical complexities $\Omega(m)$ and $\Omega(n^2)$, respectively. Using classical algorithms on the obtained sparsifier yields immediate quantum speedups for approximately solving Laplacian systems and for approximating a range of graph cut problems in essentially the same complexity. As a significantly more involved application we show how to speed up the quantum query complexity for computing exactly the edge connectivity of simple graphs. We show upper bounds on the query complexity of $\widetilde O(\sqrt{mn})$ and $\widetilde O(n^{3/2})$ in the adjacency matrix and adjacency array models, respectively. The upper bound for the adjacency matrix model is tight up to logarithmic factors, while for the adjacency array model the best quantum query lower bound we know is $\Omega(n)$.

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