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arXiv · 2504.15603

Quantum Speedup for Sampling Random Spanning Trees

Simon Apers·Minbo Gao·Zhengfeng Ji·Chenghua Liu
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Abstract

We present a quantum algorithm for sampling random spanning trees from a weighted graph in $\widetilde{O}(\sqrt{mn})$ 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 $\widetilde{O}(m)$ 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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Simon Apers, Minbo Gao, Zhengfeng Ji, Chenghua Liu. 2025-04-22. Quantum Speedup for Sampling Random Spanning Trees. https://arxiv.org/abs/2504.15603

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