Аннотация:
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. Given a set of vectors, can one deduce an effective non-negative two-body Hamiltonian with a kernel spanning this set? This can be called the 'parent Hamiltonian problem' and is the golden thread connecting the core elements of the presented research. For example, a core results (now over a decade old) solved the parent Hamiltonian problem generally for the classical case, showing that that propositional logic operations 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 strongly coupled slack spins. To solve the problem more generally, 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 showing that physically relevant two-body model Hamiltonian's have a QMA-complete 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: this is done solving the parent Hamiltonian problem for the so called, history state. Other results are also presented as they relate to ground state quantum computation and the minimisation of Hamiltonians by variational quantum circuits.
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