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

该奖项支持Sergey Bravyi、Matthew Hastings、Bruno Nachtergaele、Robert Sims、Shannon Starr、Barbara Terhal和hong - tzer Yau等七名研究人员在量子自旋系统数学理论中的三个问题群上的工作。第一个簇，局部性和利布-罗宾逊界，自旋扩散和大自旋渐近，旨在提高对量子晶格动力学的理解。第二组集中于基态性质：局部熵和纠缠的面积规律，基态以上的谱隙及其与相关函数行为的关系，以及矩阵积态近似基态的质量。第三个簇包含了计算复杂性理论中的一些问题：计算复杂性类、qma完备性、间隙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、泛函分析学家、量子计算工作者和计算机科学家最近开始发现他们各自领域中几个重要问题之间存在的密切关系。关于量子自旋哈密顿量的少数关键性质，它们产生的动力学，以及它们的基态，是解决量子自旋模型的物理行为、计算基态性质和模拟动力学的数值算法的计算效率以及量子计算理论中出现的新复杂性类等问题所需的主要成分。该项目汇集了凝聚态物理、功能分析和光谱理论、概率论和计算机科学方面的专家，以开发一个连贯的数学理论，阐明这些关键特性之间的相互关系，特别是它们与量子计算背景下量子复杂性理论新兴领域的相关性。

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.