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,\tilde \tau) queries and time. Moreover, for every integer 1 }\le \le \tau) many queries to the adjacency matrix of G. These results contrast with the classical randomized case where } \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\tilde  ) and \tilde O( \sqrt{mn + n^{3/2}), respectively, where again the edge weights are between 1 and \tau. For dense graphs we give lower bounds on the quantum query complexity of } \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) for \tau > 1 and \Omega(\tau n) for any 1 \leq \tau \leq n. Our query algorithm uses a quantum algorithm for graph sparsification } 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 .
The minimum cut problem in an undirected and weighted graph G is to find the minimum total weight of a set of edges whose removal disconnects G. We completely characterize the quantum query and time complexity of the minimum cut problem in the adjacency matrix model. If G has n vertices and edge weights at least 1 and at most \tau, we give a quantum algorithm to solve the minimum cut problem using\tilde O(n^{3/2\tau) queries and time. Moreover, for every integer 1 }\le \tau\le n we give an example of a graph G with edge weights 1 and \tau such that solving the minimum cut problem on G requires \Omega(n^{3/2\tau) many queries to the adjacency matrix of G. These results contrast with the classical randomized case where } \Omega(n^2) queries to the adjacency matrix are needed in the worst case even to decide if an unweighted graph is connected or not. In the adjacency array model, when G has m edges the classical randomized complexity of the minimum cut problem is \tilde \Theta(m). We show that the quantum query and time complexity are\tilde O(\sqrt{mn\tau ) and \tilde O( \sqrt{mn\tau + n^{3/2}), respectively, where again the edge weights are between 1 and \tau. For dense graphs we give lower bounds on the quantum query complexity of } \Omega(n ^{3/2) for \tau > 1 and \Omega(\tau n) for any 1 \leq \tau \leq n. Our query algorithm uses a quantum algorithm for graph sparsification } by Apers and de Wolf (FOCS 2020) and results on the structure of near-minimum cuts by Kawarabayashi and Thorup (STOC 2015) and Rubinstein, Schramm and Weinberg (ITCS 2018). Our time efficient implementation builds on Karger's tree packing technique (STOC 1996) .
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. 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
and results on the structure of near-minimum cuts by Kawarabayashi and Thorup (STOC 2015) and Rubinstein, Schramm and Weinberg (ITCS 2018). Our time efficient implementation builds on Karger's tree packing technique (STOC 1996)