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

TL;DR

The paper introduces controlled props by equipping a prop with a control functor, enabling control to be treated as a diagrammatic constructor. It proves that control distributes over sequential composition and satisfies a conjugation law, while not generally distributing over parallel composition, and it formalizes a complete equational theory for controllable quantum circuits (CQC) with relations acting on at most three qubits. By constructing encoding/decoding between vanilla quantum circuits (QC) and CQC, it demonstrates completeness and shows that the controlled setting allows simpler generators (global phases and Hadamard) to generate all unitaries. The work further extends the framework to points, polycontrolled props, and control in general symmetric monoidal categories, broadening the reach of diagrammatic reasoning for conditional and reversible computation. Overall, controlled props offer a compact, scalable approach to reasoning about conditioned quantum operations and other conditional processes across domains.

Abstract

Control is a predominant concept in quantum and reversible computational models. It allows to apply or not a transformation on a system, depending on the state of another system. We introduce a general framework for diagrammatic reasoning featuring control as a constructor. To do so, we provide an elementary axiomatisation of control functors, extending the standard formalism of props (symmetric monoidal categories with the natural numbers as objects) to controlled props. As an application, we show that controlled props ease diagrammatic reasoning for quantum circuits by allowing a simple complete set of relations that only involves relations acting on at most three qubits, whereas it is known that in the standard prop setting any complete axiomatisation requires relations acting on arbitrarily many qubits.

Figures (8)

• Figure 1: Complete set of relations for the graphical language of controllable quantum circuits using the controlled prop formalism. This presentation uses some shorthand notations to align with usual gate choices in quantum computing. Especially, the Z-rotation gates are in fact controlled global phases. The explicit presentation is depicted in \ref{['fig:cqcaxioms']}.
• Figure 2: Coherence laws of control functors. \ref{['eq:strenght']} is defined for any $k,n\in\mathbb{N}$ and $f\in\textbf{P}(n,n)$. \ref{['eq:controlswap']} is defined for any $n\in\mathbb{N}$ and $f\in\textbf{P}(n,n)$. \ref{['eq:swapconjugation']} is defined for any $k,p,q,\ell\in\mathbb{N}$ and $f\in\textbf{P}(k+p+q+\ell,k+p+q+\ell)$. Several wires are sometimes depicted as a single wire for simplicity.
• Figure 3: Relations $\mathcal{R}_{\textup{v}}$ for the graphical language of vanilla quantum circuits $\textbf{QC}$. The circuit $\lambda^n(\alpha)$ is defined in \ref{['fig:lambdadef']} and $\mathcal{R}_{\textup{v}}$ contains an instance of \ref{['eq:mc2pibare']} for any number of qubits $n\ge3$.
• Figure 4: Implementation of multi-controlled Z-rotations as vanilla quantum circuits.
• Figure 5: Relations $\mathcal{R}_{\textup{c}}$ for the graphical language of controllable quantum circuits $\textbf{CQC}$.

Theorems & Definitions (66)

• Example 1
• Example 2
• Example 3
• Definition 1: control functor
• Example 4
• Definition 2: controlled prop
• Definition 3: dagger controlled prop
• Definition 4: conjugation
• Example 5
• Definition 5: vanilla quantum circuits