# A Categorical Model for a Quantum Circuit Description Language (Extended Abstract)

## TL;DR

This paper introduces Proto-Quipper-M, a formal, type-safe fragment of Quipper that models quantum circuits within a categorical framework, enabling more general and rigorous descriptions of quantum computations.

## Contribution

It formalizes Proto-Quipper-M with denotational and operational semantics, extending Quipper's capabilities to a broader categorical context.

## Key findings

- Proto-Quipper-M is type-safe and sound.
- The language models parameters and state categorically.
- Proven properties include type safety, soundness, and adequacy.

## Abstract

Quipper is a practical programming language for describing families of quantum circuits. In this paper, we formalize a small, but useful fragment of Quipper called Proto-Quipper-M. Unlike its parent Quipper, this language is type-safe and has a formal denotational and operational semantics. Proto-Quipper-M is also more general than Quipper, in that it can describe families of morphisms in any symmetric monoidal category, of which quantum circuits are but one example. We design Proto-Quipper-M from the ground up, by first giving a general categorical model of parameters and state. The distinction between parameters and state is also known from hardware description languages. A parameter is a value that is known at circuit generation time, whereas a state is a value that is known at circuit execution time. After finding some interesting categorical structures in the model, we then define the programming language to fit the model. We cement the connection between the language and the model by proving type safety, soundness, and adequacy properties.