Before revision:  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 .
After revision:  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) .