Gaining new insights into the quantum-mechanical properties of various physical systems through simulations has been a central application of quantum computers since their invention in the early 1980s. As the capabilities of classical computers become insufficient for simulating scientifically interesting quantum systems and as increasingly powerful quantum computers become available, significant research effort is invested into finding and improving quantum simulation algorithms. These improvements include placing tighter bounds on the algorithmic simulation error, allowing for larger systems to be simulated with greater accuracy. In light of this trend, Childs et al. recently derived a comprehensive Trotter error theory for the very popular product formula algorithm. In this thesis, their new error analysis is applied to the Fermi- Hubbard model, which is a typical quantum simulation system because of its relevant applications, in particular in the field of high-temperature superconductors. For this system, the quantum simulation error using product formulas is shown to be about two orders of magnitude smaller than achievable with previous error bounds, demonstrating the usefulness of the work by Childs et al. This result presents a significant improvement for Fermi-Hubbard quantum simulations and suggests that similar improvements based on the new Trotter error theory are possible for other quantum systems as well. Furthermore, the results presented here are particularly relevant for near-term hardware, directly impacting current quantum simulation research.