Anomalously slow dynamics of inhomogeneous quantum annealing

Abstract

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.