the only source of errors is due to excitations generated by non-adiabatic effects in the dynamical evolution. The complexity of the adiabatic algorithms is reflected in the scaling of the minimum gap with the number of qubits. In general, AQC also requires a special form of $f(t)$ to gain a speedup as compared to classical algorithms, see Ref. [@Roland_2002] for a prominent example of an adiabatic Grover search and Ref. [@Torrentegui_2013] for a review. Moreover, optimal controlled ramps can provide additional speedups [@QOC_2011]. An alternative protocol, which is more general than Eq. , would incorporate an extra (possibly non-linear) term, such as $f(t)[1-f(t)]H_{\rm E}$, with $H_E$ being properly chosen Hamiltonian. This may have beneficial effects on the minimum gap of the system, and therefore greatly enhance the AQC performance [@non-linearAQC].

It is however clear that, especially for long annealing times, another important source of defects is related to incoherent fluctuations induced by finite temperature, or more in general by the unavoidable coupling of the system to some external environment. In this case the quantum state of the system will be mixed, described by a density matrix $\rho(t)$, satisfying a dynamical equation of the form: $$\frac{\partial \rho}{\partial t} = - \frac{i}{\hbar} 