  A unitary fusion category is called $\mathbb{Z}/2\mathbb{Z}$-quadratic if it
has a $\mathbb{Z}/2\mathbb{Z}$ group of invertible objects and one other orbit
of simple objects under the action of this group. We give a complete
classification of $\mathbb{Z}/2\mathbb{Z}$-quadratic unitary fusion categories.
The main tools for this classification are skein theory, a generalization of
Ostrik's results on formal codegrees to analyze the induction of the group
elements to the center, and a computation similar to Larson's rank-finiteness
bound for $\mathbb{Z}/3\mathbb{Z}$-near group pseudounitary fusion categories.
This last computation is contained in an appendix coauthored with attendees
from the 2014 AMS MRC on Mathematics of Quantum Phases of Matter and Quantum
Information.
