Quantum Speedup for Sampling Random Spanning Trees

• URL: http://arxiv.org/abs/2504.15603v2
• Date: Thu, 24 Apr 2025 10:30:08 GMT
• Title: Quantum Speedup for Sampling Random Spanning Trees
• Authors: Simon Apers, Minbo Gao, Zhengfeng Ji, Chenghua Liu,
• Abstract summary: We present a quantum algorithm for sampling random spanning trees from a weighted graph in $widetildeO(sqrtmn)$ time.<n>Our algorithm has sublinear runtime for dense graphs and achieves a quantum speedup over the best-known classical algorithm.
• Score: 2.356162747014486
• License: http://creativecommons.org/licenses/by/4.0/
• 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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