We present a quantum algorithm for sampling random spanning trees from a
weighted graph in $\widetilde{O}(\sqrt{mn})$ tempo, 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 $\widetilde{O}(M)$ tempo. 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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