  Accurate predictions of inclusive scattering cross sections in the linear
response regime require efficient and controllable methods to calculate the
spectral density in a strongly-correlated many-body system. In this work we
reformulate the recently proposed Gaussian Integral Transform technique in
terms of Fourier moments of the system Hamiltonian which can be computed
efficiently on a quantum computer. One of the main advantages of this framework
is that it allows for an important reduction of the computational cost by
exploiting previous knowledge about the energy moments of the spectral density.
For a simple model of medium mass nucleus like $^{40}$Ca and target energy
resolution of $1$ MeV we find an expected speed-up of $\approx 125$ times for
the calculation of the giant dipole response and of $\approx 50$ times for the
simulation of quasi-elastic electron scattering at typical momentum transfers.
