Equivalence of quantum barren plateaus to cost concentration and narrow gorges

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