It is proved that every constant-success estimator with Wasserstein--$1$ error at most $\eps$ requires $2^{\Omega(1/\eps)}$ queries, showing that the Cohen-Steiner algorithm is optimal up to constant in the exponent.
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 $2^{\Omega(1/\eps)}$ 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 $\widetilde\Omega(\eps^{-4/3})$ 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.
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