Quantum Hamiltonian Duality and Simulation

从19世纪麦克斯韦的电磁学理论，到20世纪90年代在量子引力中发现的ADS/CFT对偶，对偶性是贯穿物理学始终的深层张力。二元性的一个简单但实用的表现是模拟量子模拟实验，在该实验中，对一个系统(例如高温超导体)的物理建模的哈密顿量被直接设计并在不同的物理装置(例如光学晶格或离子陷阱)中进行研究。最近发表在《科学》杂志上的一系列论文。纳特。阿卡德。SCI。等著名期刊，将复杂性理论思想与算符代数技术相结合，发展了哈密顿模拟和对偶的数学严谨理论，并首次利用这一理论构建了能够模拟其他多体模型的通用哈密顿理论。1.扩展严格的、受复杂性理论启发的对偶性理论和哈密顿模拟，以涵盖例如依赖时间的动力学、对可用的控制类型的实验激励的约束、以及超越晶格模型的统计力学和量子多体系统。2.应用新理论拓展量子模拟的应用范围，被普遍认为是量子计算技术最早也是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.