Applying tensor decompositions to strongly correlated quantum systems

许多引人入胜的量子系统，如生物无机络合物的电子结构或发生化学反应的体系的电子和振动结构，都显示出很强的相关性。这类系统通常需要通过复杂的方法来模拟，这些方法相对于系统大小具有非常大的计算伸缩性。这种急剧的伸缩本质上是由高维张量(即大数据表)引起的。逼近它们的方法目前正在给许多科学领域带来革命性的变化，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)。矩阵乘积状态(MPS)就是一个特别的例子。在EST中，它们已经成为计算与生物相关的大型和强关联系统的基本工具。在MQD中对它们的研究较少。MPS的推广是张量网络状态(TNS)。它们克服了MPS的基本局限性，在凝聚态物理中被证明是非常有用的，但还没有在EST和MQD中被探索。这个项目的目的是利用MPS和TNS来描述EST和MQD中的强关联系统。这些领域的研究人员也开发了类似的方法，但基本上是相互孤立的。这个项目的一个中心部分是触发和利用这些领域之间的思想的交叉融合。该项目的三分之一将把从凝聚态物理和MPS的EST到基于网格的MQD的成熟的方法应用。与MQD中最先进的方法相比，MPS提供了更简单的算法结构和更低的系统规模计算伸缩性。最后的任务将是应用于具有挑战性的系统，这些系统很难用目前的方法来处理，特别是软盘和强关联分子的振动结构的计算。剩下的三分之二的项目将处理为EST开发基于网格的TN。与传统的EST方法相比，该方法在本质上是不同的，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，能够用系统大小(网格点的数量)的线性缩放来表示强关联的分子和其他量子系统。最终，这将使目前用现有方法无法实现的系统的计算成为可能。

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.