Applying tensor decompositions to strongly correlated quantum systems

许多令人着迷的量子系统，如生物无机配合物的电子结构或发生化学反应的系统的电子结构和振动结构都表现出很强的相关性。这样的系统通常需要通过复杂的方法来模拟，这些方法在系统大小方面具有非常陡峭的计算缩放。这种急剧的缩放本质上是由高维张量（即大型数据表）引起的。近似它们的方法目前正在革新许多科学领域，包括机器学习、凝聚态物理、电子结构理论（EST）和分子量子动力学（MQD）。一个特别的例子是矩阵乘积状态（MPS）。在EST中，它们已成为计算生物相关性的大型和强相关系统的基本工具。它们在MQD中较少被探索。MPS是张量网络状态（TNS）的一种推广。它们克服了MPS的基本限制，并已被证明在凝聚态物理中非常有用，但尚未在EST和MQD中进行探索。这个项目的目的是利用MPS和TNS来描述EST和MQD中的强相关系统。这些领域的研究人员已经开发出类似的方法，但在很大程度上彼此隔离。这个项目的核心部分是激发和利用这些领域之间的思想交流。项目的三分之一将应用从凝聚态物理和EST到基于网格的MQD的成熟方法。与MQD中最先进的方法相比，MPS提供了更简单的算法结构和相对于系统大小的更低的计算缩放。最后一项任务将是应用于当前方法很难处理的具有挑战性的系统，特别是软分子和强相关分子的振动结构的计算。项目剩余的三分之二将处理为EST开发基于网格的TNS。与EST的传统方法相比，拟议的技术从根本上不同，更类似于MQD中广泛使用的基于网格的方法。项目第一部分的MPS是TNS的基础。TNS具有独特的优势，能够表示具有系统大小（网格点数量）线性缩放的强相关分子和其他量子系统。最终，这将使目前现有方法无法实现的系统计算成为可能。

Many fascinating quantum systems such as the electronic structure of bioinorganic complexes or both the electronic and vibrational structure of systems undergoing a chemical reaction exhibit strong correlation. Such systems typically need to be simulated by intricate methods that have a very steep computational scaling with respect to system size. This steep scaling is essentially caused by high-dimensional tensors, that is, large tables of data. Methods to approximate them are currently revolutionizing many scientific fields, including machine learning, condensed matter physics, electronic structure theory (EST) and molecular quantum dynamics (MQD). One particular example are matrix product states (MPS). In EST, they have become an essential tool to compute large and strongly correlated systems of biological relevance. They are less explored in MQD.A generalization of MPS are tensor network states (TNS). They overcome fundamental limitations of MPS and have proven to be very useful in condensed matter physics but have not yet been explored neither in EST nor in MQD.The aim of this project is to utilize both MPS and TNS for describing strongly correlated systems both in EST and MQD. Researchers in these areas have developed similar methods, but largely isolated from each other. A central part of this project is to trigger and to utilize cross-fertilizations of ideas between these areas.One third of the project will apply the well-established methodology from condensed matter physics and EST for MPS to grid-based MQD. Compared to state-of-the-art methods in MQD, MPS offer a simpler algorithmic structure and a lower computational scaling with respect to system size. The final task will be applications to challenging systems that are very difficult to handle with current methods, especially the computation of the vibrational structure of floppy and strongly correlated molecules.The remaining two thirds of the project will deal with developing grid-based TNS for EST. Compared to conventional approaches in EST, the proposed technique is fundamentally different and resembles more the grid-based methods extensively used in MQD. MPS of the first part of the project are the foundations of TNS. TNS have the distinctive advantage of being able to represent strongly correlated molecules and other quantum systems with a linear scaling of system size (number of grid points). Ultimately, this will enable computations of systems that are currently beyond reach with existing methods.