Two dimensional gauge invariant tensor networks

对处于平衡态和非平衡态的相关量子多体系统的研究--尽管是我们理解自然的核心--仍然是量子物理学的一大挑战。可用的分析和数值方法在许多情况下是非常有限的，并且仅允许描述感兴趣的物理学的近似版本。在最近的十年中，张量网络方法开辟了新的可能性，现在是模拟一维相关量子动力学的标准工具。然而，二维多体量子系统动力学在大多数有趣的环境中仍然无法进行数值模拟。提供新的数值算法来解决这一挑战，将为量子技术、量子场论、凝聚态和临界现象等不同物理领域带来光明。基于我最近为一维系统介绍的一些发展-格点规范张量网络，最优控制理论和分层张量网络-以及高性能计算，该项目旨在采取前所未有的步骤：-二维格点规范理论的数值模拟方法的发展;- 研究当今阻碍量子技术发展的开放问题以及我们对受抑自旋系统性质的理解（量子二聚体模型，价键态），拓扑模型，和高能量二维模型的晶格描述方面的量子电动力学和色动力学。该项目将为在未知领域的研究铺平道路，对相关多体量子物理学的基本方面产生影响。该项目的结果也将作为第一代量子模拟器的高度非平凡的基准，并且-由于其实现的极端复杂性-用于其验证。

The study of correlated quantum many-body systems in and out of equilibrium - despite being at the heart of our understanding of Nature - remains one of the big challenge of quantum physics. Analytical and numerical methods available are in many cases strongly limited and allow to describe only an approximated version of the physics of interests. In the lat decade, tensor network methods opened new possibilities and are now a standard tool to simulate one-dimensional correlated quantum dynamics. However, two-dimensional many-body quantum system dynamics are still out of reach of numerical simulations in most interesting settings. Providing novel numerical algorithms to solve this challenge, would shade light into different fields of physics as quantum technologies, quantum field theories, condensed matter and critical phenomena. Building on some recent developments I have introduced for one-dimensional systems -lattice gauge tensor networks, optimal control theory, and hierarchical tensor networks- together with high- performance computing, this project aims to take unprecedented steps towards: - the development of numerical simulations methods for two-dimensional lattice gauge theories; - the study of open problems that nowadays hinder the development of quantum technologies and our understanding of the properties of frustrated spin systems (quantum dimer models, valence bond states), topological models, and high-energy two dimensional models on a lattice describing aspects of quantum electrodynamics and chromodynamics. This project will pave the way to investigations in unchartered territory with impact on fundamental aspects of correlated many-body quantum physics. The results of this project will serve also as highly non trivial benchmarks for the first generation of quantum simulators, and - due to the extreme complexity of their implementation - for their verification.