Obstacles to inhomogeneous quantum annealing

Inhomogeneous quantum annealing (IQA) has been shown to circumvent the first-order phase transitions hindering conventional quantum annealing (QA) by turning off transverse fields one at a time, rather than simultaneously. Here, we explicitly analyze IQA dynamics, revealing that IQA, due to spin-freezing events, must proceed more slowly than what the phase diagram would generically suggest. More consequentially, the resulting block-diagonal Hamiltonians that appear mid-protocol in non-mean-field models, relevant for hard optimization problems, feature gaps that vanish exactly (not just exponentially). Both phenomena stem from the simple fact that a spin's magnetization would remain conserved when its field is removed. We conclude by quantifying IQA's performance compared to QA across a thousand realizations of hard problems.