Classical and quantum spectral density estimation under local graph access

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 $2<sup>{O(1/\varepsilon)}$ local queries to the graph. In this paper, we prove that every constant-success estimator with Wasserstein--$1$ error at most $\eps$ requires $2<sup>{Ω(1/\eps)}$ 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 $\widetilde O(\eps<sup>{-3})$-query algorithm estimating the spectral density with Wasserstein-1 error at most $\eps$. Finally, we prove a $\widetildeΩ(\eps<sup>{-4/3})$ 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.