Quantum Speedup for Sampling Random Spanning Trees

2025-04-22

AI Summary

This work addresses quantum sampling of uniformly random spanning trees in weighted graphs. Classically, the optimal algorithm runs in $widetilde{O}(m)$ time, where $m$ is the number of edges. We present the first quantum algorithm with runtime $widetilde{O}(sqrt{mn})$, where $n$ is the number of vertices—achieving an $Omega(sqrt{n})$ quantum speedup for dense graphs. Our approach integrates, for the first time, large-step classical random walks, quantum graph sparsification, a no-replacement variant of Hamoudi’s multistate preparation, amplitude estimation, and mixing-time optimization. We prove the complexity is tight: a matching quantum lower bound shows our algorithm is optimal up to a $mathrm{polylog}$ factor. This constitutes the first complete solution for spanning tree sampling that simultaneously achieves provable quantum speedup, asymptotically tight analysis, and practically structured design.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Sampling/Simulation-based Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Quantum algorithm for sampling random spanning trees
• Achieves quantum speedup over classical methods
• Optimal algorithm for dense graph sampling

Innovation

• Quantum algorithm for spanning tree sampling
• Combines classical and quantum techniques
• Achieves sublinear runtime for dense graphs