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arXiv:2609.39457

A Quantum Scaling Algorithm for Maximum-Weight Perfect Matching in General Graphs

Kourosh Mirsohi, Sandy Irani, Michael T. Goodrich

cs.DSquant-phSubmitted Sep 30, 2026• Announced Oct 1, 2026• 0 citations
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Abstract

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 \( \widetilde{O}(n m^{2/3}\log W) \) 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 \( \widetilde{O}(m\sqrt n \log W) \) in the dense regime, where $m\ge n^{3/2}$. 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.

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40 pages, 7 figures, 3 tables

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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 (
\widetilde{O}(n m^{2/3}\log W) ) time, where WWW is an upper bound on the magnitude of the edge weights. This is an improvement over the best known classical combinatorial bound of (
\widetilde{O}(m\sqrt n \log W) ) in the dense regime, where m≥n3/2m\ge n^{3/2}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.
We present a quantum algorithm for sampling random spanning trees from a weighted graph in Õ(√{mn}) time, where n and m denote the number of vertices and edges, respectively. Our algorithm has sublinear runtime for dense graphs and achieves a quantum speedup over the best-known classical algorithm, which runs in Õ(m) time. The approach carefully combines, on one hand, a classical method based on "large-step" random walks for reduced mixing time and, on the other hand, quantum algorithmic techniques, including quantum graph sparsification and a sampling-without-replacement variant of Hamoudi’s multiple-state preparation. We also establish a matching lower bound, proving the optimality of our algorithm up to polylogarithmic factors. These results highlight the potential of quantum computing in accelerating fundamental graph sampling problems.
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Abstract Combinatorial optimization problems over graphs, such as the traveling salesman problem, longest path problem, and maximum independent set problem, are well-known for being computationally costly, some even NP-hard problems. In this paper, we propose a general quantum algorithm framework searching for approximate solutions to combinatorial optimization problems with linear objective functions. Our framework provides APIs (application programming interfaces) that enable developers to encode weighted graph structures onto quantum circuits and utilize variational algorithms to generate approximate solutions. One key advantage of our framework is that it allows developers to design new graph algorithms for the graph problem represented as linear combinations of edge weights without requiring expertise in quantum programming. Besides, it only uses a logarithmic level of quantum bit scale, making our framework work on quantum computers with limited physical resources. Our experimental results demonstrate that our framework can provide good approximations for the traveling salesman problem compared to current quantum algorithm.
In this paper, we consider the parameterized quantum query complexity for graph problems. We design parameterized quantum query algorithms for k-vertex cover and k-matching problems, and present lower bounds on the parameterized quantum query complexity. Then, we show that our quantum query algorithms are optimal up to a constant factor when the parameters are small. Our main results are as follows. Parameterized quantum query complexity of vertex cover. In the k-vertex cover problem, we are given an undirected graph G with n vertices and an integer k, and the objective is to determine whether G has a vertex cover of size at most k. We show that the quantum query complexity of the k-vertex cover problem is O(√kn + k^{3/2}√n) in the adjacency matrix model. For the design of the quantum query algorithm, we use the method of kernelization, a well-known tool for the design of parameterized classical algorithms, combined with Grover’s search. Parameterized quantum query complexity of matching. In the k-matching problem, we are given an undirected graph G with n vertices and an integer k, and the objective is to determine whether G has a matching of size at least k. We show that the quantum query complexity of the k-matching problem is O(√kn + k²) in the adjacency matrix model. We obtain this upper bound by using Grover’s search carefully and analyzing the number of Grover’s searches by making use of potential functions. We also show that the quantum query complexity of the maximum matching problem is O(√pn + p²) where p is the size of the maximum matching. For small p, it improves known bounds Õ(n^{3/2}) for bipartite graphs [Blikstad-v.d.Brand-Efron-Mukhopadhyay-Nanongkai, FOCS 2022] and O(n^{7/4}) for general graphs [Kimmel-Witter, WADS 2021]. Lower bounds on parameterized quantum query complexity. We also present lower bounds on the quantum query complexities of the k-vertex cover and k-matching problems. The lower bounds prove the optimality of the above parameterized quantum query algorithms up to a constant factor when k is small. Indeed, the quantum query complexities of the k-vertex cover and k-matching problems are both Θ(√k n) when k = O(√n) and k = O(n^{2/3}), respectively.
Inspired by the Elitzur-Vaidman bomb testing problem [arXiv:hep-th/9305002], we introduce a new query complexity model, which we call bomb query complexity B(f)B(f)B(f). We investigate its relationship with the usual quantum query complexity Q(f)Q(f)Q(f), and show that B(f)=Θ(Q(f)2)B(f)=Θ(Q(f)^2)B(f)=Θ(Q(f)2). This result gives a new method to upper bound the quantum query complexity: we give a method of finding bomb query algorithms from classical algorithms, which then provide nonconstructive upper bounds on Q(f)=Θ(B(f))Q(f)=Θ(\sqrt{B(f)})Q(f)=Θ(B(f)​). We subsequently were able to give explicit quantum algorithms matching our upper bound method. We apply this method on the single-source shortest paths problem on unweighted graphs, obtaining an algorithm with O(n1.5)O(n^{1.5})O(n1.5) quantum query complexity, improving the best known algorithm of O(n1.5log⁡n)O(n^{1.5}\sqrt{\log n})O(n1.5logn​) [arXiv:quant-ph/0606127]. Applying this method to the maximum bipartite matching problem gives an O(n1.75)O(n^{1.75})O(n1.75) algorithm, improving the best known trivial O(n2)O(n^2)O(n2) upper bound.
The fastest known classical algorithm deciding the kkk-colorability of nnn-vertex graph requires running time Ω(2n)Ω(2^n)Ω(2n) for k≥5k\ge 5k≥5. In this work, we present an exponential-space quantum algorithm computing the chromatic number with running time O(1.9140n)O(1.9140^n)O(1.9140n) using quantum random access memory (QRAM). Our approach is based on Ambainis et al's quantum dynamic programming with applications of Grover's search to branching algorithms. We also present a polynomial-space quantum algorithm not using QRAM for the graph 202020-coloring problem with running time O(1.9575n)O(1.9575^n)O(1.9575n). In the polynomial-space quantum algorithm, we essentially show (4−ε)n(4-ε)^n(4−ε)n-time classical algorithms that can be improved quadratically by Grover's search.