A quantum complexity lower bound from differential geometry

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