The computational power of many-body systems

Abstract

Many-body systems are well known throughout physics to be hard problems to exactly solve, but much of this is folklore resulting from the lack of an analytic solution to these systems. This thesis attempts to classify the complexity inherent in many of these systems, and give quantitative results for why the problems are hard. In particular, we analyze the many-particle system corresponding to a multi-particle quantum walk, showing that the time evolution of such systems on a polynomial sized graph is universal for quantum computation, and thus determining how a particular state evolves is as hard as an arbitrary quantum computation. We then analyze the ground energy properties of related systems, showing that for bosons, bounding the ground energy of the same Hamiltonian with a fixed number of particles is QMA-complete. Similar techniques provide a novel proof that quantum walk is universal for quantum computing, and constructs a QMA-complete problem that does not reference quantum mechanics.