Quantum speed-ups have been obtained for many fundamental graph problems, including most variants of matching. A notable exception, however, is the maximum-weight perfect matching (MWPM) problem in general graphs with integer edge weights, which is arguably the most challenging variant of matching. We present a quantum algorithm for MWPM in general graphs that runs in O(nm2/3logW) time, where W is an upper bound on the magnitude of the edge weights. This is an improvement over the best known classical combinatorial bound of O(mnlogW) in the dense regime, where m≥n3/2. To the best of our knowledge, this is the first quantum algorithm to obtain an asymptotic improvement over the best classical combinatorial algorithm for the MWPM problem in general graphs. The running time of our method accounts for QRAM initialization and access overheads up to polylogarithmic factors, as well as all classical updates to the data structures. At a high level, our algorithm is based on a classical framework due to Duan, Pettie, and Su, but our algorithm requires replacing certain classical tasks with quantum methods, alternative analysis of classical procedures, and the use of alternative data structures that can be implemented effectively in the QRAM model.
If a quantum walk starting on a vertex tends to stay at home, then that vertex is said to be sedentary. We prove that almost all planar graphs and almost all trees contain at least two sedentary vertices for any assignment of edge weights --- a result that suggests vertex sedentariness is a common phenomenon in trees and planar graphs. For weighted bipartite graphs, we show that a vertex is not sedentary whenever 0 does not belong to its eigenvalue support. Consequently, each vertex in a nonsingular weighted bipartite graph is not sedentary, a stark contrast to weighted trees and weighted planar graphs. A corollary of this result is that every vertex in a bipartite graph with a unique perfect matching is not sedentary for any assignment of edge weights. We also construct new families of weighted bipartite graphs with sedentary vertices using the bipartite double, subdivision operation, and corona product. Finally, we show that unweighted paths and unweighted even cycles contain no sedentary vertices.
Abstract We investigate quantum and nonsignaling generalizations of perfect matchings in graphs using nonlocal games. Specifically, we introduce nonlocal games that test for L-perfect matchings in bipartite graphs, perfect matchings in general graphs and hypergraphs, and fractional perfect matchings. Our definitions come from the fact that these games are classical property tests for the corresponding matching conditions. We use the existence of perfect quantum and nonsignaling strategies for these games to define quantum and nonsignaling versions of perfect matchings. Finally, we provide characterizations of when graphs exhibit these extended properties: For nonsignaling matchings, we give a complete combinatorial characterization. In particular, a graph has a nonsignaling perfect matching if and only if it admits a fractional perfect matching that has bounded value on triangles. In bipartite graphs, the nonsignaling L-perfect matching property is achieved exactly when the left component of the graph can be split into two disjoint subgraphs: one with a classical L-perfect matching and another with left-degree 2. In the quantum setting, we show that complete graphs Kn K n with odd n≥7 n ≥ 7 have quantum perfect matchings. We prove that a graph has a quantum perfect matching if and only if the quantum independence number of its line graph is maximal, extending a classical relationship between perfect matchings and line graph independence numbers. For bipartite graphs, we establish that the L-perfect matching game does not exhibit quantum pseudotelepathy, but we characterize the quantum advantage for complete bipartite graphs Kn,2 K n , 2 . Additionally, we prove that deciding quantum perfect matchings in hypergraphs is undecidable and leaves open the question of its complexity in graphs.