  Optimizing parameterized quantum circuits (PQCs) is the leading approach to
make use of near-term quantum computers. However, very little is known about
the cost function landscape for PQCs, which hinders progress towards
quantum-aware optimizers. In this work, we investigate the connection between
three different landscape features that have been observed for PQCs: (1)
exponentially vanishing gradients (called barren plateaus), (2) exponential
cost concentration about the mean, and (3) the exponential narrowness of minina
(called narrow gorges). We analytically prove that these three phenomena occur
together, i.e., when one occurs then so do the other two. A key implication of
this result is that one can numerically diagnose barren plateaus via cost
differences rather than via the computationally more expensive gradients. More
broadly, our work shows that quantum mechanics rules out certain cost
landscapes (which otherwise would be mathematically possible), and hence our
results are interesting from a quantum foundations perspective.
