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.

翻译：简单图形的边缘连通性是最小的边缘, 其去除使图形断开。 我们研究在对称矩阵和对称阵列模型中, 以对称矩阵和对称阵列模型计算简单图形的边缘连通性, 以美元为单位计算对称的边缘连通性。 对于经典随机算法, $\ Omega (n ⁇ 2) 美元 和 $\\ Omega(m) 的下限, 这些模型分别是已知的, 这些模型的微值算法在查询复杂度方面是最佳的。 相反, 我们用美元(n ⁇ 3/2}) 和 $(tilde) 在对称对称对称矩阵和对称阵列阵列矩阵模型中, 最差的算法是 $\\\ 美元(n) 。 我们的对等离值的对等值的对等值的对等值的对等值直径直径直径对等值, 将Ormal m 的对等的对等直径直径直径直径对等的对等数 。 我们的对等直径直径直径直径对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对数, 对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等,对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对等的对