Spectral density estimation with the Gaussian Integral Transform

Understand

The spectral density operator $\hat{\rho}(\omega)=\delta(\omega-\hat{H})$ 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 $\Delta$ and error $\epsilon$ using $\mathcal{O}\left(\sqrt{\log\left(1/\epsilon\right)\left(\log\left(1/\Delta\right)+\log\left(1/\epsilon\right)\right)}/\Delta\right)$ operations.
• This is achieved without using expensive approximations to the time-evolution operator but exploiting instead qubitization to implement an approximate Gaussian Integral Transform (GIT) of the spectral density.
• We also describe appropriate error metrics to assess the quality of spectral function approximations more generally.