One rig to control them all

TL;DR

This paper addresses how to capture computational control in a purely structural way by showing that control is exactly the structure of rig categories. It introduces eight control equations and a universal construction that, for any base prop $\mathbf{P}$, yields a controlled prop $\mathbf{cP}$ whose semantics form the free rig category on $\mathbf{P}$. The authors prove universality and coherence via Gray induction and demonstrate significant consequences: $\mathbf{cX}$ is isomorphic to $\mathbf{Perm}_2$, providing universal control for reversible circuits, and, with a Hadamard gate or a single $\sqrt{X}$ gate, yield universal controlled quantum theories with complete equational frameworks. The framework further enables syntactic derivations of decompositions and optimizations (e.g., Sleator-Weinfurter) and connects to complete equational theories for quantum circuits across gate sets, highlighting the structural nature of controlled computation.

Abstract

We introduce a theory for computational control, consisting of eight naturally interpretable equations. Adding these to a prop of base circuits constructs controlled circuits, borne out in examples of reversible Boolean circuits and quantum circuits. We prove that this syntactic construction semantically corresponds to taking the free rig category on the base prop.

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