Characterizing entangling quantum dynamics

Abstract

When can one quantum system be used to simulate another? What aspects of a quantum system restrict its capacity to simulate other systems? Understanding the limitations and possibilities of such simulations is a key issue at the intersection of physics and computer science. In this thesis we determine what physical systems can be simulated by finite-dimensional quantum systems with entangling Hamiltonians and complete local unitary control. By an entangling Hamiltonian we mean each of the system's subsystems are coupled, either directly or indirectly. This very broad class of Hamiltonians is applicable to a wide range of important physical systems in Nature.Our key motivation for studying such simulation problems is quantum computing. It can be shown that a locally controllable Hamiltonian system must be entangling if it is to be used to make a quantum computer. By characterizing the simulation properties of entangling Hamiltonian systems, we identify necessary (and in some cases sufficient) properties for such systems to be used in the implementation of a quantum computer.We perform our categorization of entangling Hamiltonian systems by determining the Lie algebras they can generate with local unitary control. We find that the key to this categorization is given by simple properties of the structure of the system Hamiltonian. Our main result is a complete characterization of all entangling Hamiltonian systems based on the set of Hamiltonians that they can generate with local unitary control.