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Determining properties of ground states of spin Hamiltonians remains a topicof central relevance connecting disciplines of mathematical, theoretical andapplied physics. In the last few decades, ground state properties of physicalsystems have been increasingly considered as computational resources. Thisthesis develops parts of the mathematical apparatus to create (program) groundstates relevant for quantum and classical computation. The core findingspresented in this thesis (now over a decade old) including that (i) logicoperations (gates) can be embedded into the low-energy sector of Ising spinswhereas three (and higher) body Ising interaction terms can be mimicked throughthe minimisation of 2- and 1-body Ising terms yet require the introduction ofslack spins; (ii) Perturbation theory gadgets enable the emulation ofinteractions not present in a given Hamiltonian, e.g.~$YY$ interactions can berealized from $ZZ$, $XX$, the thesis contains a result from 2007 showing thatphysically relevant two-body model Hamiltonian's have a QMA-hard ground stateenergy decision problem. Merged with other results, this established that thesemodels provide a universal resource for ground state quantum computation. Morerecent findings include the proof that an idealised version of the contemporaryvariational approach to quantum algorithms enables a universal model of quantumcomputation. Other related results are also presented as they relate to groundstate quantum computation and the minimisation of Hamiltonians by quantumcircuits. The topics covered include: Ising model reductions, stochastic versusquantum processes on graphs, quantum gates and circuits as tensor networks,variational quantum algorithms and Hamiltonian gadgets.

On the mathematical structure of quantum models of computation based on Hamiltonian minimisation