AI 中文总结
该研究针对局部图访问下的谱密度估计问题，证明了经典算法的最优性，提出了量子局部访问模型下的多项式级复杂度算法并给出下界，解决了相关开放问题。
AI 中文摘要
我们研究局部访问模型下无向图归一化邻接矩阵的谱密度估计问题。此前Cohen-Steiner等人[KDD 2018]提出了一种算法，用于在Wasserstein-1距离下进行ε近似谱密度估计，该算法需要对图进行2^{O(1/ε)}次局部查询。在本文中，我们证明了每一个成功概率为常数、且Wasserstein-1误差不超过ε的估计器，都需要2^{Ω(1/ε)}次查询，这表明Cohen-Steiner算法在指数项的常数因子范围内是最优的，从而解决了Jin等人[COLT 2023]和Peng等人[COLT 2026]留下的开放问题。随后我们转向量子局部访问模型，给出了一个查询复杂度为Õ(ε^{-3})、且Wasserstein-1误差不超过ε的谱密度估计算法，最后证明了当图足够大时，量子查询的下界为Õ(ε^{-4/3})。因此，量子局部访问模型将对ε的依赖从指数级变为多项式级。
英文摘要
We study spectral density estimation for the normalized adjacency matrix of an unweighted graph under local access model. Previously, Cohen-Steiner et al. [KDD 2018] proposed an algorithm for $\varepsilon$-approximate spectral density estimation in the Wasserstein-1 distance, using $2^{O(1/\varepsilon)}$ local queries to the graph. In this paper, we prove that every constant-success estimator with Wasserstein--$1$ error at most $\eps$ requires $2^{Ω(1/\eps)}$ queries, showing that the Cohen-Steiner algorithm is optimal up to constant in the exponent. This resolves the open problem left by previous researches Jin et al. [COLT 2023] and Peng et al. [COLT 2026]. We then turn to quantum local access model. We give an $\widetilde O(\eps^{-3})$-query algorithm estimating the spectral density with Wasserstein-1 error at most $\eps$. Finally, we prove a $\widetildeΩ(\eps^{-4/3})$ quantum lower bound when the graph is sufficiently large. As a result, quantum local access model changes the dependence on $\eps$ from exponential to polynomial.
