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We introduce and study the adiabatic dynamics of free-fermion models subjectto a local Lindblad bath and in the presence of a time-dependent Hamiltonian.The merit of these models is that they can be solved exactly, and will help usto study the interplay between non-adiabatic transitions and dissipation inmany-body quantum systems. After the adiabatic evolution, we evaluate theexcess energy (average value of the Hamiltonian) as a measure of the deviationfrom reaching the target final ground state. We compute the excess energy in avariety of different situations, where the nature of the bath and theHamiltonian is modified. We find a robust evidence of the fact that an optimalworking time for the quantum annealing protocol emerges as a result of thecompetition between the non-adiabatic effects and the dissipative processes. Wecompare these results with matrix-product-operator simulations of an Isingsystem and show that the phenomenology we found applies also for this morerealistic case.

Dissipation in adiabatic quantum computers: Lessons from an exactly solvable model