  In this thesis, I investigate aspects of local Hamiltonians in quantum
computing. First, I focus on the Adiabatic Quantum Computing model, based on
evolution with a time dependent Hamiltonian. I show that to succeed using AQC,
the Hamiltonian involved must have local structure, which leads to a result
about eigenvalue gaps from information theory. I also improve results about
simulating quantum circuits with AQC. Second, I look at classically simulating
time evolution with local Hamiltonians and finding their ground state
properties. I give a numerical method for finding the ground state of
translationally invariant Hamiltonians on an infinite tree. This method is
based on imaginary time evolution within the Matrix Product State ansatz, and
uses a new method for bringing the state back to the ansatz after each
imaginary time step. I then use it to investigate the phase transition in the
transverse field Ising model on the Bethe lattice. Third, I focus on locally
constrained quantum problems Local Hamiltonian and Quantum Satisfiability and
prove several new results about their complexity. Finally, I define a
Hamiltonian Quantum Cellular Automaton, a continuous-time model of computation
which doesn't require control during the computation process, only preparation
of product initial states. I construct two of these, showing that time
evolution with a simple, local, translationally invariant and time-independent
Hamiltonian can be used to simulate quantum circuits.
