Quantum Speedup for Sampling Random Spanning Trees
Simon Apers, Minbo Gao, Zhengfeng Ji, Chenghua Liu
TL;DR
The paper addresses the problem of efficiently sampling random spanning trees from weighted graphs in the quantum setting. It introduces QRST, a quantum algorithm that uses a large-step down-up walk combined with domain sparsification under isotropy and quantum multi-sampling to achieve a near-optimal runtime of $\widetilde{O}(\sqrt{mn})$ (up to polylog factors) and an $\varepsilon$-accurate distribution from $\mathcal{W}_G$. The contributions include a matching lower bound $\Omega(\sqrt{mn})$ for quantum query complexity, a detailed construction of a quantum resistance oracle, and novel quantum subroutines such as $\mathsf{QIsotropicSample}$ and a Hamoudi-based sampling-without-replacement technique. This work demonstrates a concrete quantum speedup for a fundamental graph sampling task and points to future directions in applying quantum techniques to determinantal point processes and related sampling problems.
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.