The Complexity Geometry of a Single Qubit

Abstract

The computational complexity of a quantum state quantifies how hard it is to make. `Complexity geometry', first proposed by Nielsen, is an approach to defining computational complexity using the tools of differential geometry. Here we demonstrate many of the attractive features of complexity geometry using the example of a single qubit, which turns out to be rich enough to be illustrative but simple enough to be illuminating.

45 pages, 8 figures, 1 qubit. v2: added extraneous appendix about optimal shapes for tunnels bored through Earth