FRG: Collaborative Research: Quantum SpinSystems. Theory and Applications in Quantum Computation

该奖项支持谢尔盖·布拉维、马修·黑斯廷斯、布鲁诺·纳赫特盖勒、罗伯特·西姆斯、香农·斯塔尔、芭芭拉·特哈尔和洪泽佑七位研究人员在量子自旋系统数学理论中的三个问题上所做的工作。第一个是团簇，局域性和Lieb-Robinson边界，自旋扩散和大自旋渐近性，目的是提高对量子晶格动力学的理解。第二类集中于基态性质：局域熵和纠缠的面积定律，基态上方的光谱带隙及其与关联函数行为的关系，以及矩阵积态对基态的逼近质量。第三类群包含计算复杂性理论中的一些问题：计算复杂性类、QMA完备性、有间隙的哈密顿量与复杂性之间的联系，以及随机哈密顿量的计算能力，所有这些都与量子自旋系统有关。凝聚态物理学家、数学物理学家、泛函分析师、量子计算工作者和计算机科学家最近开始发现各自领域中的几个重要问题之间存在着密切的关系。关于量子自旋哈密顿量、它们产生的动力学和它们的基态的一小部分关键性质是解决关于量子自旋模型的物理行为、关于计算基态性质和模拟动力学的数值算法的计算效率以及关于量子计算理论中出现的新的复杂类别的问题所需的主要成分。这个项目汇集了凝聚态物理、泛函分析和光谱理论、概率理论和计算机科学方面的专家，以开发一种连贯的数学理论，阐明这些关键性质之间的相互关系，特别是它们与量子计算背景下的量子复杂性理论这一新兴领域的相关性。

This award supports the work of a group of seven researchers, Sergey Bravyi, Matthew Hastings, Bruno Nachtergaele, Robert Sims, Shannon Starr, Barbara Terhal, and Horng-Tzer Yau, on three clusters of problems in the mathematical theory of quantum spin systems. The first cluster, locality and Lieb-Robinson bounds, spin diffusion, and large-spin asymptotics, is aimed at improving understanding of quantum lattice dynamics. The second cluster focuses on ground state properties: area laws for the local entropy and entanglement, the spectral gap above the ground state and its relation with the behavior of correlation functions, and the quality of approximation of ground states by matrix product states. The third cluster contains a number of questions in computational complexity theory: computational complexity classes, QMA-completeness, the connection between gapped Hamiltonians and complexity, and the computational power of stoquastic Hamiltonians, all of which relate to quantum spin systems.Condensed matter physicists, mathematical physicists, functional analysts, workers in quantum computation, and computer scientists recently have begun to discover the close relationships that exist between several of the important questions in their respective fields. A small number of key properties about quantum spin Hamiltonians, the dynamics they generate, and their ground states are the main ingredients needed to address questions about the physical behavior of quantum spin models, about the computational efficiency of numerical algorithms to compute ground state properties and simulate dynamics, and about new complexity classes that are emerging in the theory of quantum computation. This project brings together experts in condensed matter physics, functional analysis and spectral theory, probability theory, and computer science to develop a coherent mathematical theory that clarifies the interrelationships of these key properties and, in particular, their relevance for the emerging field of quantum complexity theory in the context of quantum computation.