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We argue that a complete description of quantum annealing (QA) implementedwith continuous variables must take into account the non-adiabaticAharonov-Anandan geometric phase that arises when the system Hamiltonianchanges during the anneal. We show that this geometric effect leads to theappearance of non-stoquastic terms in the effective quantum Ising Hamiltoniansthat are typically used to describe QA with flux-qubits. We explicitlydemonstrate the effect of these geometric interactions when QA is performedwith a system of one and two coupled flux-qubits. The realization ofnon-stoquastic Hamiltonians has important implications from a computationalcomplexity perspective, since it is believed that in many cases QA withstoquastic Hamiltonians can be efficiently simulated via classical algorithmssuch as Quantum Monte Carlo. It is well-known that the direct implementation ofnon-stoquastic interactions with flux-qubits is particularly challenging. Ourresults suggest an alternative path for the implementation of non-stoquasticinteractions via geometric phases that can be exploited for computationalpurposes.

Non-Stoquastic Interactions in Quantum Annealing via the Aharonov-Anandan Phase