# Categorical Quantum Circuits

*Ville Bergholm and Jacob Biamonte*

### Abstract

This paper takes steps to bring the fields of category theory and quantum information science into closer contact. Specifically, we present an explicit representation of dagger compact closed categories in terms of an extended form of quantum circuit diagrams which we have used to produce results new to both areas. We map the string diagram calculus previously used in the categorical axiomatization of quantum mechanics onto quantum circuits. This approach enables novel, powerful ways of manipulating and simplifying circuits of systems of arbitrary dimension, while remaining within a model that is directly and easily applicable to problems stated in the language of quantum information science. The circuit diagrams themselves now become morphisms in a category, making quantum circuits a special case of a much more general mathematical framework. This approach introduces a new set of tools which can be used to manipulate and simplify quantum networks in novel ways. We have strived to present these ideas in a way that does not require a category theory background to ease the reader into the subject. All the main proofs we give use nothing but standard linear algebra techniques. Our specific approach has further applications in applying category theory and related ideas to tensor network simulation.