Quantum Hamiltonian Duality and Simulation

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.