发表机构
Institute of Software, Chinese Academy of Sciences; University of Chinese Academy of Sciences; School of Computer Science, Shanghai Jiao Tong University; Department of Computer Science and Technology, Tsinghua University（中国科学院软件研究所; 中国科学院大学; 上海交通大学计算机学院; 清华大学计算机科学与技术系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对具有局部分解结构的对数凹采样问题，提出量子算法，将局部查询复杂度从经典最优的线性依赖改进为条件数的平方根依赖，证明局部结构是量子加速的关键资源。
AI 中文摘要
对于凸函数 $f \colon \mathbb{R}^d \to \mathbb{R}$，从与 $e^{-f(x)}$ 成正比的分布中采样的问题称为对数凹采样。在许多实际场景中，函数 $f(x)$ 可分解为局部形式 $f(x) = \sum_{a=1}^R \psi_a(x_{S_a})$。本文考虑使用局部查询（即对每个子句 $\psi_a(\cdot)$ 的求值和梯度查询）进行对数凹采样，这比直接查询 $f(x)$ 本身在计算上便宜得多。我们证明，如果每个坐标仅出现在少量子句中，则存在一种量子算法，用于强对数凹采样，其局部查询复杂度为 $\widetilde{O}(\sqrt{\kappa}d)$，其中 $\kappa$ 是条件数。这改进了 Ascolani、Lavenant 和 Zanella（Ann. Probab. 2026）先前的最佳经典结果 $\widetilde{O}(\kappa d)$，以及 Childs 等人（NeurIPS 2022）隐含的量子结果 $\widetilde{O}(\sqrt{\kappa} d^2)$。我们的量子采样器适用于统计计算和机器学习中一大类局部结构模型，代表性示例包括高斯马尔可夫随机场、有限元潜在高斯模型和稀疏广义线性模型。这些结果表明，局部结构不仅仅是实现细节，而是高维采样的量子算法资源。
英文摘要
For a convex function $f \colon \mathbb{R}^d \to \mathbb{R}$, the problem of sampling from a distribution proportional to $e^{-f(x)}$ is called log-concave sampling. In many practical scenarios, the function $f(x)$ turns out to admit a local decomposition $f(x) = \sum_{a=1}^R ψ_a(x_{S_a})$. In this paper, we consider log-concave sampling using local queries, i.e., evaluation and gradient queries to each clause $ψ_a(\cdot)$, which can be computationally much cheaper than the queries to $f(x)$ itself. We show that if each coordinate appears in only a small number of clauses, there is a quantum algorithm for strongly log-concave sampling using $\widetilde{O}(\sqrtκd)$ local queries, where $κ$ is the condition number. This improves the prior best classical result $\widetilde{O}(κd)$ due to Ascolani, Lavenant, and Zanella (Ann. Probab. 2026) and the quantum result $\widetilde{O}(\sqrtκ d^2)$ implied by Childs et al. (NeurIPS 2022). Our quantum sampler applies to a broad class of locally structured models from statistical computing and machine learning, with representative examples including Gaussian Markov random fields, finite-element latent Gaussian models, and sparse generalized linear models. These results demonstrate that local structure is not merely an implementation detail, but a quantum algorithmic resource for high-dimensional sampling.
