A Quantum Complexity Lowerbound from Differential Geometry

TL;DR

This paper uses differential geometry, specifically the Bishop-Gromov bound, to establish tight lower bounds on quantum complexity, showing it is exponentially large for typical unitaries across various metrics.

Contribution

It introduces a novel approach applying differential geometry to derive the first broad, tight lower bounds on quantum complexity for a wide class of metrics.

Findings

Quantum complexity is exponentially large for typical unitaries.

The bounds are tighter than all previously known lower bounds.

The method applies to a broad class of complexity geometry metrics.

Abstract

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.