The main focus of this proposal is research into improving Matrix Product State Methods for quantum many-body physics. In the first phase of the project, we numerically studied so called Spin-Holstein model (Knörzer et al., 2022) as an example of a relevant and challenging quantum many-body system. In particular, we improved upon existing tensor network state (TNS) methods by applying a recently proposed integration of Gaussian state methods into TNS. (Nüsseler et al. 2021) We reproduced the proposed approach, adapted it to Spin-Holstein models, and towards the end of the first phase were able to further improve the method going beyond Gaussian states. The computational resources of this project will allow us to benchmark these novel methods prior to then publishing and sharing the developed code with the community. In addition to this, the resources granted may be used towards research in improved numerical methods for the study of Causal Fermion Systems (building on arXiv:2201.06382), hardware-optimised design of star-to-chain transformations for analogue quantum simulation or numerical simulations related to adiabatic quantum computing.