Geometry of abstraction in quantum computation

TL;DR

This paper explores the categorical semantics of quantum algorithms, analyzing function abstraction and classical interfaces, and demonstrates implementing quantum algorithms using abelian groups and relations.

Contribution

It advances the understanding of quantum computation's categorical structure and provides a method to implement quantum algorithms with algebraic relations.

Findings

Categorical semantics clarify quantum function abstraction.

Quantum algorithms can be implemented with abelian groups and relations.

Insights into resources needed for quantum algorithms.

Abstract

Quantum algorithms are sequences of abstract operations, performed on non-existent computers. They are in obvious need of categorical semantics. We present some steps in this direction, following earlier contributions of Abramsky, Coecke and Selinger. In particular, we analyze function abstraction in quantum computation, which turns out to characterize its classical interfaces. Some quantum algorithms provide feasible solutions of important hard problems, such as factoring and discrete log (which are the building blocks of modern cryptography). It is of a great practical interest to precisely characterize the computational resources needed to execute such quantum algorithms. There are many ideas how to build a quantum computer. Can we prove some necessary conditions? Categorical semantics help with such questions. We show how to implement an important family of quantum algorithms using just abelian groups and relations.