The Category CNOT
Robin Cockett (University of Calgary), Cole Comfort (University of, Calgary), Priyaa Srinivasan (University of Calgary)

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.
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