A quantum complexity lower bound from differential geometry

Abstract

Differential geometry has long found applications in physics in general relativity and related areas. More recently, it was proposed by Nielsen that the tools of differential geometry, when applied to the unitary group, might be used to bound the complexity of quantum operations. The Bishop–Gromov bound—a cousin of the focusing lemmas used to prove the Penrose–Hawking black hole singularity theorems—is a differential geometry result that gives an upper bound on the rate of growth of the volume of geodesic balls in terms of the Ricci curvature. Here I apply the Bishop–Gromov bound to Nielsen’s complexity geometry to prove lower bounds on the quantum complexity of a typical unitary. For a broad class of models, the typical complexity is shown to be exponentially large in the number of qubits. This technique gives results that are tighter than all known lower bounds in the literature, as well as establishing lower bounds for a much broader class of complexity geometry metrics than has hitherto been bounded. This method thus realizes the original vision of Nielsen, which was to apply the tools of differential geometry to study quantum complexity.