Quantum Hamiltonian Duality and Simulation

从19世纪麦克斯韦的电磁学理论，到20世纪90年代量子引力中AdS/CFT对偶的发现，对偶是贯穿物理学的一种深刻的张力。对偶的一个简单但实际的表现是模拟量子模拟实验，其中一个模拟一个系统（例如高温超导体）的物理的哈密顿量是直接在不同的物理设置（例如光学晶格或离子阱）中设计和研究的。最近发表在《科学》杂志上的一系列论文。和其他著名期刊，将复杂性理论思想与算子代数技术结合起来，发展了一个数学上严格的哈密顿模拟和对偶理论，并利用这个理论第一次构造了“通用哈密顿”，能够模拟其他所有多体模型。这个博士项目的目的是双重的。1. 扩展严格的，复杂性理论启发的对偶和哈密顿模拟理论，以涵盖例如时间依赖动力学，实验动机约束的控制类型，以及统计力学和超越晶格模型的量子多体系统。2. 应用新理论扩大量子模拟应用的范围，被广泛认为是量子计算技术最早和最重要的实际应用之一。

Duality is a deep strain running through physics, from Maxwell's 19th century theory of electromagnetism, to the 1990s discovery of AdS/CFT duality in quantum gravity. A simple but practical manifestation of duality is analogue quantum simulation experiments, where a Hamiltonian modelling the physics of one system (e.g. high-temperature superconductors) is directly engineered and studied in a different physical setup (e.g. optical lattices or ion traps). A recent series of papers published in Science, Proc. Nat. Acad. Sci. and other prominent journals, combined complexity theory ideas with operator algebra techniques to develop a mathematically rigorous theory of Hamiltonian simulation and duality, and used this to construct for the first time "universal Hamiltonians", capable of simulating every other many-body model.The aim of this PhD project is two-fold. 1. To extend the rigorous, complexity-theory-inspired theory of duality and Hamiltonian simulation to encompass e.g. time-dependent dynamics, experimentally-motivated constraints on the types of control available, and statistical mechanics and quantum many-body systems beyond lattice models. 2. To apply the new theory to expand the reach of quantum simulation applications, widely expected to be one of the earliest and most important practical applications of quantum computing technology.