Direct comparison of quantum and simulated annealing on a fully connected Ising ferromagnet

We compare the performance of quantum annealing (QA, through Schrödinger dynamics) and simulated annealing (SA, through a classical master equation) on the p-spin infinite range ferromagnetic Ising model, by slowly driving the system across its equilibrium, quantum or classical, phase transition. When the phase transition is second order (p=2, the familiar two-spin Ising interaction) SA shows a remarkable exponential speed-up over QA. For a first-order phase transition (p>=3, i.e., with multispin Ising interactions), in contrast, the classical annealing dynamics appears to remain stuck in the disordered phase, while we have clear evidence that QA shows a residual energy which decreases towards zero when the total annealing time ? increases, albeit in a rather slow (logarithmic) fashion. This is one of the rare examples where a limited quantum speedup, a speedup by QA over SA, has been shown to exist by direct solutions of the Schrödinger and master equations in combination with a nonequilibrium Landau-Zener analysis. We also analyze the imaginary-time QA dynamics of the model, finding a 1/?2 behavior for all finite values of p, as predicted by the adiabatic theorem of quantum mechanics. The Grover-search limit p(odd)=? is also discussed.