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arXiv:2508.07007

A Quantum Walk-Driven Algorithm for the Minimum Spanning Tree Problem under a Maximal Degree Constraint

F. S. Luiz, F. F. Fanchini, Victor Hugo C. de Albuquerque, João Paulo Papa, M. C. de Oliveira

Published Aug 9, 2025• 0 citations
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Abstract

We present a novel quantum walk-based approach to solve the Minimum Spanning Tree (MST) problem under a maximal degree constraint (MDC). By recasting the classical MST problem as a quantum walk on a graph, where vertices are encoded as quantum states and edge weights are inverted to define a modified Hamiltonian, we demonstrate that the quantum evolution naturally selects the MST by maximizing the cumulative transition probability (and thus the Shannon entropy) over the spanning tree. Our method, termed Quantum Kruskal with MDC, significantly reduces the quantum resource requirement to $\mathcal{O}(\log N)$ qubits while retaining a competitive classical computational complexity. Numerical experiments on fully connected graphs up to $10^4$ vertices confirm that, particularly for MDC values exceeding $4$, the algorithm delivers MSTs with optimal or near-optimal total weights. When MDC values are less or equal to $4$, some instances achieve a suboptimal solution, still outperforming several established classical algorithms. These results open promising perspectives for hybrid quantum-classical solutions in large-scale graph optimization.

DOI
10.48550/arxiv.2508.07007

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Quantum annealing (QA) is a different technology from gate-model quantum computing. This research proposes a novel technique for solving the Minimum Spanning Tree (MST) (or, the minimum weight spanning tree) problem on a quantum annealer. The problem is of interest due to its applications in clustering, unsupervised learning, network design, and image processing to name a few. The advent of quantum cloud computing has provided access to quantum computing tools, previously unavailable to the general community. D-Wave systems recently released cloud access to their quantum annealer type hardware, which this project leverages to provide a novel solution method to the MST problem.
Constraint Satisfaction Problems (CSPs) are a fundamental class of computational problems prevalent in various fields, including artificial intelligence, optimization, and cryptography. Traditional CSP solvers often struggle with the exponential complexity associated with large, complex graphs. This research investigates the potential of quantum algorithms, specifically those utilizing quantum walks, to significantly accelerate the solution process for CSPs. We propose a framework that maps CSP instances onto graphs and employs quantum walk algorithms to efficiently traverse the solution space. The core of our approach leverages the quantum superposition principle, enabling the simultaneous exploration of multiple potential solutions. We demonstrate, through theoretical analysis and algorithmic design, that quantum walks can offer a substantial speedup compared to classical algorithms for certain CSP instances, particularly those with high connectivity and large solution spaces. This work contributes a novel approach to tackling computationally intensive CSPs by integrating quantum computing principles with graph traversal techniques, representing a potentially transformative advancement in solving complex optimization challenges.
We present a quantum algorithm that generates a uniform superposition over the spanning trees of a graph -- also known as a q-sample -- using only a sub-linear number of queries to the graph. This goes beyond previous work that focuses on generating classical samples, including a recent quantum algorithm by Apers, Gao, Ji, and Liu [ICALP 2025]. For an nnn-vertex, mmm-edge graph, our algorithm admits a pre-processing step of O~(mn+m1−δ)\tilde{O}(\sqrt{mn} + m^{1-δ})O~(mn​+m1−δ), after which each q-sample can be generated in O~(n1+2δ)\tilde{O}(n^{1+2δ})O~(n1+2δ) for any δδδ. Alternatively, kkk independent q-samples can be generated at a total cost of O~(kmn)\tilde{O}(\sqrt{kmn})O~(kmn​). We also prove a matching lower bound up to logarithmic factors, showing that our algorithm is essentially optimal. Further tradeoffs are given in the paper. In comparison, the optimal classical algorithms (Anari, Liu and Vuong [FOCS 2022]) need an O~(m)\tilde{O}(m)O~(m) pre-processing step, after which each sample costs O~(n)\tilde{O}(n)O~(n) operations. These results yield faster quantum walk-based algorithms for counting spanning trees or finding a marked one. Our result is obtained via quantum walk sampling over a sequence of slowly-changing Markov chains. Each chain is an isotropized up-down walk that mixes rapidly to the spanning tree distribution of the input graph. A key ingredient is an amortized data structure supporting fast implementation of the associated quantum walk operators throughout the sequence. This data structure maintains access to a spectral sparsifier and a leverage score sampler that evolve with the underlying graph.