The power of density: Suppression of quantum phase transitions via dense driver Hamiltonians in adiabatic quantum computation

In the context of adiabatic quantum computation (AQC), it has been argued that first order quantum phase transitions (QPTs) due to localisation phenomena will always cause adiabatic quantum computation (AQC) to fail by exponentially decreasing the minimal spectral gap of the Hamiltonian along the annealing path. However this notion has been subject to some debate in the community, since more recent findings suggest the existence of methods to avoid this by carefully designing the involved Hamiltonians. It remains a challenge to formulate a comprehensive theory on the effect of the various parameters and the conditions under which QPTs make the AQC algorithm fail. In this work we investigate the conditions under which localisation causes first order QPTs. As a consequence of this analysis and using methods from spectral graph theory, we here examine both analytically and numerically the role of the connectivity of the driver Hamiltonian in the mitigation of such effects in different AQC algorithms and show that in the limiting case of full connectivity, first order QPTs due to localisation are avoided entirely.