Spectral-density estimation with the Gaussian integral transform

The spectral-density operator ρ(ω)=δ(ω-Ĥ) plays a central role in linear response theory as its expectation value, the dynamical response function, can be used to compute scattering cross sections. In this work, we describe a near optimal quantum algorithm providing an approximation to the spectral density with energy resolution Δ and error ϵ using Olog21/ϵlog21/Δ+log21/ϵ/Δ operations. This is achieved without using expensive approximations to the time-evolution operator, but instead exploiting qubitization to implement an approximate Gaussian integral transform of the spectral density. We also describe appropriate error metrics to assess the quality of the spectral function approximations more generally.