Quantum query complexity of edge connectivity

The edge connectivity of a simple graph is the least number of edges whose removal disconnects the graph. We study the quantum query complexity of computing the edge connectivity of a simple graph with $n$ vertices and $m$ edges in both the adjacency matrix and adjacency array models. For classical randomized algorithms, $\Omega(n^2)$ and $\Omega(m)$ lower bounds are known for these models, respectively, showing that the trivial algorithm is optimal in terms of query complexity for both cases. In contrast, we show upper bounds of $\tilde O(n^{3/2})$ and $\tilde O(\sqrt{mn})$ on the quantum query complexity of computing edge connectivity in the adjacency matrix and adjacency array models, respectively. Our upper bound is tight up to logarithmic factors for the adjacency matrix model, while for the adjacency array model the best quantum query lower bound we know is $\Omega(n)$. Our upper bounds follow by combining a lemma on the structure of near-minimum cuts due to Rubinstein, Schramm and Weinberg (ITCS 2018) with a quantum sparsification algorithm by Apers and de Wolf (FOCS 2020). For a weighted graph, instead of edge connectivity the relevant notion is the weight of a minimum cut, i.e. the minimum total weight of a set of edges whose removal disconnects the graph. For weighted graphs we show that in the worst case $\Omega(n^2)$ queries can be needed to compute the weight of a minimum cut by a quantum algorithm in both the adjacency matrix and adjacency array models, thus no quantum speedup is possible in this case.