# Diagrammatic Reasoning with Control as a Constructor, Applications to Quantum Circuits

## TL;DR

This paper introduces a diagrammatic reasoning framework with control as a constructor, enabling simplified and complete reasoning about quantum circuits using controlled props with relations on at most three qubits.

## Contribution

It extends the formalism of props to controlled props, providing an elementary axiomatisation and simplifying quantum circuit reasoning.

## Key findings

- Controlled props enable diagrammatic reasoning with relations on at most three qubits.
- Standard props require relations acting on arbitrarily many qubits for completeness.
- The framework simplifies reasoning about quantum circuits and their control structures.

## Abstract

Control is a fundamental concept in quantum and reversible computational models. It enables the conditional application of a transformation to a system, depending on the state of another system. We introduce a general framework for diagrammatic reasoning featuring control as a constructor. To this end, we provide an elementary axiomatisation of control functors, extending the standard formalism of props to controlled props. As an application, we show that controlled props facilitate diagrammatic reasoning for quantum circuits by introducing a simple and complete set of relations involving at most three qubits, whereas in the standard prop setting any complete axiomatisation necessarily requires relations acting on arbitrarily many qubits.

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Source: https://tomesphere.com/paper/2508.21756