Magnetic molecules, modelled as finite-size spin systems, are test-beds for quantum phenomena1 and could constitute key elements in future spintronics devices2,3,4,5, long-lasting nanoscale memories6 or noise-resilient quantum computing platforms7,8,9,10. Inelastic neutron scattering is the technique of choice to probe them, characterizing molecular eigenstates on atomic scales11,12,13,14. However, although large magnetic molecules can be controllably synthesized15,16,17,18, simulating their dynamics and interpreting spectroscopic measurements is challenging because of the exponential scaling of the required resources on a classical computer. Here, we show that quantum computers19,20,21,22 have the potential to efficiently extract dynamical correlations and the associated magnetic neutron cross-section by simulating prototypical spin systems on a quantum hardware22. We identify the main gate errors and show the potential scalability of our approach. The synergy between developments in neutron scattering and quantum processors will help design spin clusters for future applications.