  Determining properties of ground states of spin Hamiltonians remains a topic
of central relevance connecting disciplines of mathematical, theoretical and
applied physics. In the last few decades, ground state properties of physical
systems have been increasingly considered as computational resources. This
thesis develops parts of the mathematical apparatus to create (program) ground
states relevant for quantum and classical computation. The core findings
presented in this thesis (now over a decade old) including that (i) logic
operations (gates) can be embedded into the low-energy sector of Ising spins
whereas three (and higher) body Ising interaction terms can be mimicked through
the minimisation of 2- and 1-body Ising terms yet require the introduction of
slack spins; (ii) Perturbation theory gadgets enable the emulation of
interactions not present in a given Hamiltonian, e.g.~$YY$ interactions can be
realized from $ZZ$, $XX$, the thesis contains a result from 2007 showing that
physically relevant two-body model Hamiltonian's have a QMA-hard ground state
energy decision problem. Merged with other results, this established that these
models provide a universal resource for ground state quantum computation. More
recent findings include the proof that an idealised version of the contemporary
variational approach to quantum algorithms enables a universal model of quantum
computation. Other related results are also presented as they relate to ground
state quantum computation and the minimisation of Hamiltonians by quantum
circuits. The topics covered include: Ising model reductions, stochastic versus
quantum processes on graphs, quantum gates and circuits as tensor networks,
variational quantum algorithms and Hamiltonian gadgets.
