Inhomogeneous quantum annealing (IQA), in which transverse fields are turned off one by one rather than simultaneously, has been proposed as an effective way to avoid the first-order phase transitions that impede conventional quantum annealing (QA). Here we explicitly study the dynamics of IQA, rather than merely the thermodynamics, and find that it is appreciably slower than the phase diagram would suggest. Interestingly, this slowdown manifests both when IQA succeeds in circumventing phase transitions and when it fails. Even in the absence of transitions, such as for the mean-field models that have been analyzed previously, IQA is slower than expected by a factor of the number of spins
N. More significantly, we show that in non-mean-field models, first-order transitions are likely to be quite common, and the gap at such transitions is not merely exponential in
N but exactly zero. Thus IQA cannot reach the ground state on any timescale. Both of these results can be understood through the simple observation that a spin's magnetization becomes conserved once its field is turned off during the IQA protocol.
We investigate quantum annealing (QA) with antiferromagnetic transverse interactions for the generalized Hopfield model with k -body interactions. The goal is to study the effectiveness of antiferromagnetic interactions, which were shown to help us avoid problematic first-order quantum phase transitions in pure ferromagnetic systems, in random systems. We estimate the efficiency of QA by analyzing phase diagrams for two cases where the number of embedded patterns is finite or extensively large. The phase diagrams of the model with finite patterns show that there exist annealing paths that avoid first-order transitions at least for . The same is true for the extensive case with k = 4 and 5. In contrast, it is impossible to avoid first-order transitions for the case of finite patterns with k = 3 and the case of extensive number of patterns with k = 2 and 3. The spin-glass phase hampers the QA process in the case of k = 2 and extensive patterns. These results indicate that QA with antiferromagnetic transverse interactions is efficient also for certain random spin systems.
We compare the performance of quantum annealing (QA, through Schr\"odinger 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 ($p\ensuremath{\ge}3$, 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 $\ensuremath{\tau}$ 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 Schr\"odinger and master equations in combination with a nonequilibrium Landau-Zener analysis. We also analyze the imaginary-time QA dynamics of the model, finding a $1/{\ensuremath{\tau}}^{2}$ behavior for all finite values of
p, as predicted by the adiabatic theorem of quantum mechanics. The Grover-search limit $p(\mathrm{odd})=\ensuremath{\infty}$ is also discussed.