The Category CNOT

TL;DR

This paper characterizes the category generated by CNOT and related gates, establishing its algebraic structure and equivalences to categories of affine isomorphisms over Z2 vector spaces.

Contribution

It provides a complete set of identities for the CNOT category and proves its equivalence to the category of affine partial isomorphisms over Z2 vector spaces.

Findings

CNOT category is a discrete inverse category.

CNOT is equivalent to the category of affine partial isomorphisms over Z2 vector spaces.

The paper establishes a complete set of identities for CNOT.

Abstract

We exhibit a complete set of identities for CNOT, the symmetric monoidal category generated by the controlled-not gate, the swap gate, and the computational ancillae. We prove that CNOT is a discrete inverse category. Moreover, we prove that CNOT is equivalent to the category of partial isomorphisms of finitely-generated non-empty commutative torsors of characteristic 2. Equivalently this is the category of affine partial isomorphisms between finite-dimensional Z2 vector spaces.