# Free quantum computing

## TL;DR

This paper introduces a new framework for quantum computing that simplifies its foundations and clarifies its relationship with classical computing.

## Contribution

A novel discrete and symbolic framework for quantum computing is proposed, replacing traditional continuous models with discrete axioms and a category-theoretical model.

## Key findings

- The framework isolates quantum advantage in the ability to take well-behaved square roots.
- The free model allows combinatorial optimization and automated verification of quantum computations.
- The model is computationally universal and can be linked to various quantum hardware platforms.

## Abstract

Quantum computing holds great promise, but its foundations and the source of its advantages remain conceptually obscure. We develop a framework that contains no extraneous mathematical assumptions and clearly separates what is truly quantum from what is just classical computing in disguise. Instead of relying on the infinite precision of continuous complex numbers, this symbolic approach uses a small finite number of discrete building blocks that reflect physical implementations. Unlike traditional models, this model supports purely symbolic combinatorial reasoning, enabling the use of powerful classical computer science techniques. This framework is just as effective as traditional ones, and offers a rigorous, simpler foundation to understand and engineer quantum computation.

Quantum computing improves substantially on known classical algorithms for various important problems, but the nature of the relationship between quantum and classical computing is not yet fully understood. This relationship can be clarified by free models, that add to classical computing just enough physical principles to represent quantum computing and no more. Here, we develop an axiomatization of quantum computing that replaces the standard continuous postulates with a small number of discrete equations, as well as a free model that replaces the standard linear-algebraic model with a category-theoretical one. The axioms and model are based on reversible classical computing, isolate quantum advantage in the ability to take certain well-behaved square roots, and link to various quantum computing hardware platforms. This approach allows combinatorial optimization, including brute force computer search, to optimize quantum computations. The free model may be interpreted as a programming language for quantum computers, that has the same expressivity and computational universality as the standard model, but additionally allows automated verification and reasoning.