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Abstract:

A large class of optimisation problems can be mapped to the Ising model where all details are encoded in the coupling of spins. The task of the original mathematical optimisation is then equivalent to finding the ground state of the corresponding spin system which can be achieved via quantum annealing relying on the adiabatic theorem. Some of the inherent disadvantages of this procedure can be alleviated or resolved using a stochastic approach, and by coupling to the external environment. We show that careful engineering of the system-bath coupling at an individual spin level can further improve annealing

Abstract:

Contextuality is a nonclassical feature of quantum systems---exhibited by data that is produced empirically or theoretically---which in the realm of sheaf theory is characterised by local consistency but global inconsistency. A large part of this thesis is concerned with studying how this signature of nonclassicality is apparent also when measurements are embedded in some causal structure, and so motivating the study of causal contextuality. We begin with temporal correlations, which occur when measurements are performed sequentially on the system, and in which the definition of nonclassicality becomes sensitive to memory resources with which the classical system is equipped. For certain types of such memory, we show that there exists a map from the temporal setup to a (appropriately defined) contextuality setup, such that every nonclassical temporal empirical model satisfying no-signalling constraints consistent with the memory function corresponds to a contextual empirical model on this constructed scenario---one can view this also as a simulation of a subset of the temporal correlations by the contextuality setup. The existence of such a map allows us to apply a result from Vorob'ev in order to say, for any temporal setup and choice memory function, whether nonclassical correlations can arise. We then study causal setups by employing the notion of strategy from game semantics. We in particular show how `playing off' Nature strategies, corresponding to adaptive hidden variables, against Experimenter strategies, which may also be adaptive, realises the classical correlations of certain causal setups from the literature. We show that adaptivity on the side of the Experimenter, by reducing the sets of measurements empirical data is obtained over, can remove the inconsistencies that are imperative for the observation of contextuality. In the second part of the thesis, we study spin Hamiltonians on random graphs, focusing on exact descriptions in the thermodynamic limit. By utilising the graphon, which is the limit object of a dense random graphs sequence, we are able to derive analytical results for certain graphons and certain choice of Hamiltonian. Our overarching result is that the equilibrium physics in the thermodynamic limit is described by a set of coupled equations containing the graphon, and which describes product, \ie unentangled, states