Harnessing the geometry of tensor networks for the simulation of complex quantum matter

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Tensor network schemes have proven their ability to describe complex quantum systems numerically and analytically. They have originated from basic yet powerful numerical techniques which have matured into a body of methods and formalisms capturing the relevant degrees of freedom of complex quantum systems in terms of tensor networks. And yet, it seems fair to say that a comprehensive understanding is still lacking. There is a flurry of activity of devising new numerical methods, but the mathematical understanding of the underlying structures can so far not keep pace. An example where a good understanding of the underlying mathematical structures has been reached is that of fixed rank matrix product states. And even here, not all lessons seem to have been drawn from this understanding when it comes to optimizing numerical techniques based on these insights.This project sets out for gaining an understanding and improving algorithms for the description of static and dynamic properties of especially non-local quantum systems, based on a deepened applied mathematical understanding of the underlying tensor network structures. We plan to do this by bringing methods, tools and insights from algebraic and differential geometry as well as graph theory into a fresh context and apply them to tensor network methods. In this project, we hence suggest a concerted effort between physics and mathematics, and bring together expertise from both branches of research. From these considerations, we plan to significantly improve algorithms for the description of static and dynamic properties, especially of non-local quantum systems as encountered in the context of quantum chemistry. Equipped with those tools, we envision to further merge tensor network schemes with different complementary numerical approaches to quantum systems and explore the potential of the resulting combined methods.