Match the quantum query upper and lower bounds for spectral density estimation

Match the quantum query-complexity upper and lower bounds for estimating the spectral density of a graph under the quantum local access model, improving the gap between the current \widetilde O(\varepsilon^{-3}) upper bound and the \widetilde\Omega(\varepsilon^{-4/3}) lower bound.

Background

The paper studies spectral density estimation for the normalized adjacency matrix of a graph using local graph queries. In the classical local access model, it establishes an exponential query complexity of 2^{\Theta(1/\varepsilon)}, matching the prior upper bound up to constants in the exponent.

For the quantum local access model, the paper develops an estimator using \widetilde O(\varepsilon^{-3}) quantum graph queries and proves a \widetilde\Omega(\varepsilon^{-4/3}) lower bound. The authors explicitly identify closing this polynomial gap—either by improving the algorithmic upper bound, strengthening the lower bound, or both—as an unresolved problem.