  Two main challenges preventing efficient training of variational quantum
algorithms and quantum machine learning models are local minima and barren
plateaus. Typically, barren plateaus are associated with deep circuits, while
shallow circuits have been shown to suffer from suboptimal local minima. We
point out a simple mechanism that creates exponentially many poor local minima
specifically in the barren plateau regime. These local minima are trivial
solutions, optimizing only a few terms in the loss function, leaving the rest
on their barren plateaus. More precisely, we show the existence of approximate
local minima, optimizing a single loss term, and conjecture the existence of
exact local minima, optimizing only a logarithmic fraction of all loss function
terms. One implication of our findings is that simply yielding large gradients
is not sufficient to render an initialization strategy a meaningful solution to
the barren plateau problem.
