  The Bishop-Gromov bound -- a cousin of the focusing lemmas that Hawking and
Penrose used to prove their black hole singularity theorems -- is a
differential geometry result that upperbounds the rate of growth of volume of
geodesic balls in terms of the Ricci curvature. In this paper, I apply the
Bishop-Gromov bound to Nielsen's complexity geometry to prove lowerbounds on
the quantum complexity of a typical unitary. For a broad class of penalty
schedules, the typical complexity is shown to be exponentially large in the
number of qubits. This technique gives results that are tighter than all known
lowerbounds in the literature, as well as establishing lowerbounds for a much
broader class of complexity geometry metrics than has hitherto been bounded.
For some metrics, I prove these lowerbounds are tight. This method realizes the
original vision of Nielsen, which was to apply the tools of differential
geometry to study quantum complexity.
