# The Complexity Geometry of a Single Qubit

## TL;DR

This paper explores the complexity geometry of a single qubit, demonstrating its features and illustrating how differential geometry can be used to understand quantum state complexity.

## Contribution

It provides a detailed analysis of complexity geometry applied to a single qubit, highlighting its illustrative power and potential for understanding quantum complexity.

## Key findings

- Complexity geometry captures key features of quantum state complexity.
- Single qubit complexity geometry is rich and illustrative.
- Differential geometry offers valuable tools for quantum complexity analysis.

## 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.