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.
