Classical and quantum spectral density estimation under local graph access

2026-08-23

AI Summary

本文解决了无权图归一化邻接矩阵的谱密度估计问题，通过局部访问模型证明了经典算法的最优性，并提出了量子算法将误差依赖从指数级改进到多项式级。

Abstract

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.

Problem

• Spectral Density Estimation
• Local Access Model
• Wasserstein-1 Distance
• Query Complexity

Innovation

• spectral density estimation
• local access model
• quantum algorithm
• Wasserstein-1 distance
• query complexity