Figure 5: Simulated quantum annealing in the open system.

Using a Monte Carlo wave-function description averaged over 100 runs, we simulate the annealing process with N=4 ions in the presence of dephasing and spontaneous emission. With a noise rate Γ=0.03 J_rms ≈120 Hz and a total annealing time T=1 ms, we have on average one jump per run, . We plot expectation values of the spins and total number of phonons, produced by the coupling, as a function of time. Here, we have operated in the ferromagnetic regime between the second and third phonon resonance, δ=2π × 159 kHz. Choosing Ω=2π × 50 kHz, ω_recoil=2π × 15 kHz, ω_z=2π × 876 kHz, B_max=50 J_rms≈200 kHz, and  kHz, we are able to perform very fast annealing,  ms, with high fidelity. After a time T=1 ms, the time evolution without noise has converged to =(0.97, 0.70, −0.65, −0.85), correctly reproducing the classical pattern (++−−).