Correlations and measures of entanglement in ground state wavefunctions of relativistic quantum field theories are spatially localized over length scales set by the mass of the lightest particle. We utilize this localization to design digital quantum circuits for preparing the ground states of free lattice scalar quantum field theories. Controlled rotations that are exponentially localized in their position-space extent are found to provide exponentially convergent wavefunction fidelity. These angles scale with the classical two-point correlation function, as opposed to the more localized mutual information or the hyper-localized negativity. We anticipate that further investigations will uncover quantum circuit designs with controlled rotations dictated by the measures of entanglement. This work is expected to impact quantum simulations of systems of importance to nuclear physics, high-energy physics, and basic energy sciences research.