Figure 1: Illustration of thermal noise effects near an anticrossing.

An avoided crossing (anticrossing) of two lowest-energy eigenstates, and , with a small minimum energy gap, g_min, separated from other eigenstates by energy δE. Passing through the anticrossing very quickly swaps the probabilities of the ground and first excited states (blue and green dotted arrows), leaving the probabilities of and unchanged. Thus, for a closed-system starting in , the final ground-state probability, P_b, would be vanishingly small. However, with an environment at T>0, thermal transitions can excite the system beforehand and relax it afterward (red arrows), the net effect of which is to increase P_b (green arrow). Single-qubit tunnelling amplitudes are significantly larger before the anticrossing (see Fig. 2b), making thermal excitations earlier in the annealing process much more likely than relaxation later. If T≳δE/k_B, higher excited states would also be occupied, reducing P_b; so a peak in P_b is expected at T=T_peak∼δE/k_B.