In this talk, we highlight our group's recent developments on computing the Compton amplitude in a lattice approach. We briefly discuss how to access the Compton amplitude directly via the second-order Feynman-Hellmann theorem. As an application, we compute the nucleon Compton tensor across a range of photon momenta at an unphysical quark mass. This enables us to study the $Q^2$ dependence of the low moments of the nucleon structure functions in a lattice calculation for the first time. We discuss possible further applications of this approach.