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

Cet article explore les excursions dans le temps et leurs implications.

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