发表机构
LaBRI, Univ. Bordeaux, CNRS UMR 5800（拉布里实验室，波尔多大学，法国国家科学研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一种量子算法，利用摊销数据结构在缓慢变化的马尔可夫链上进行量子游走采样，以次线性查询生成图的生成树均匀叠加态，并证明其最优性，优于经典算法。
AI 中文摘要
我们提出了一种量子算法，该算法生成图的所有生成树的均匀叠加态——也称为q-样本——仅需对图进行次线性次数的查询。这超越了先前专注于生成经典样本的工作，包括Apers、Gao、Ji和Liu [ICALP 2025]最近提出的量子算法。对于具有$n$个顶点、$m$条边的图，我们的算法允许一个预处理步骤，其复杂度为$\tilde{O}(\sqrt{mn} + m^{1-\delta})$，之后每个q-样本可以在$\tilde{O}(n^{1+2\delta})$时间内生成，其中$\delta$为任意值。或者，$k$个独立的q-样本可以以总成本$\tilde{O}(\sqrt{kmn})$生成。我们还证明了匹配的下界（直到对数因子），表明我们的算法本质上是最优的。论文中还给出了进一步的权衡。相比之下，最优的经典算法（Anari、Liu和Vuong [FOCS 2022]）需要$\tilde{O}(m)$的预处理步骤，之后每个样本花费$\tilde{O}(n)$次操作。这些结果为计数生成树或寻找标记生成树提供了更快的基于量子游走的算法。我们的结果是通过在一系列缓慢变化的马尔可夫链上进行量子游走采样获得的。每条链都是一个各向同性的上下游走，它快速混合到输入图的生成树分布。一个关键成分是一种摊销数据结构，支持在整个序列中快速实现相关的量子游走算子。该数据结构维护对谱稀疏化和杠杆分数采样器的访问，这些采样器随底层图演化。
英文摘要
We present a quantum algorithm that generates a uniform superposition over the spanning trees of a graph -- also known as a q-sample -- using only a sub-linear number of queries to the graph. This goes beyond previous work that focuses on generating classical samples, including a recent quantum algorithm by Apers, Gao, Ji, and Liu [ICALP 2025]. For an $n$-vertex, $m$-edge graph, our algorithm admits a pre-processing step of $\tilde{O}(\sqrt{mn} + m^{1-δ})$, after which each q-sample can be generated in $\tilde{O}(n^{1+2δ})$ for any $δ$. Alternatively, $k$ independent q-samples can be generated at a total cost of $\tilde{O}(\sqrt{kmn})$. We also prove a matching lower bound up to logarithmic factors, showing that our algorithm is essentially optimal. Further tradeoffs are given in the paper. In comparison, the optimal classical algorithms (Anari, Liu and Vuong [FOCS 2022]) need an $\tilde{O}(m)$ pre-processing step, after which each sample costs $\tilde{O}(n)$ operations. These results yield faster quantum walk-based algorithms for counting spanning trees or finding a marked one. Our result is obtained via quantum walk sampling over a sequence of slowly-changing Markov chains. Each chain is an isotropized up-down walk that mixes rapidly to the spanning tree distribution of the input graph. A key ingredient is an amortized data structure supporting fast implementation of the associated quantum walk operators throughout the sequence. This data structure maintains access to a spectral sparsifier and a leverage score sampler that evolve with the underlying graph.
