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

张量网络方案已经证明了它们在数值和解析上描述复杂量子系统的能力。它们起源于基本而强大的数值技术，这些技术已经成熟为一系列方法和形式，以张量网络的形式捕捉复杂量子系统的相关自由度。然而，似乎可以公平地说，仍然缺乏全面的理解。在设计新的数值方法方面有一系列的活动，但对基本结构的数学理解到目前为止还跟不上步伐。一个已经达到对基本数学结构的很好理解的例子是固定秩矩阵积状态。即使在这里，当涉及到基于这些见解的优化数值技术时，似乎并不是所有的教训都来自于这种理解。这个项目旨在获得对描述特别是非局域量子系统的静态和动态性质的理解和改进算法，基于对潜在张量网络结构的深化应用数学理解。我们计划通过将代数和微分几何以及图论的方法、工具和见解引入一个全新的背景并将它们应用到张量网络方法中来实现这一点。因此，在这个项目中，我们建议物理学和数学之间的协调努力，并将两个研究分支的专业知识结合在一起。基于这些考虑，我们计划显著改进描述静态和动态性质的算法，特别是在量子化学背景下遇到的非局域量子系统。配备了这些工具，我们设想进一步将张量网络格式与不同的互补数值方法合并到量子系统中，并探索由此产生的组合方法的潜力。

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.