Categories of quantum cpos

TL;DR

This paper introduces quantum cpos, a noncommutative generalization of classical cpos, using discrete quantization, to develop categorical models for quantum programming languages.

Contribution

It presents the concept of quantum cpos via discrete quantization, extending domain theory to quantum structures for modeling quantum programming languages.

Findings

Quantum cpos share categorical properties with classical cpos.

Quantum cpos are suitable for constructing models of quantum programming languages.

Quantum cpos may serve as the foundation for quantum domain theory.

Abstract

This paper unites two research lines. The first involves finding categorical models of quantum programming languages and their type systems. The second line concerns the program of quantization of mathematical structures, which amounts to finding noncommutative generalizations (also called quantum generalizations) of these structures. Using a quantization method called discrete quantization, which essentially amounts to the internalization of structures in a category of von Neumann algebras and quantum relations, we find a noncommutative generalization of $\omega$-complete partial orders (cpos), called quantum cpos. Cpos are central in domain theory, and are widely used to construct categorical models of programming languages. We show that quantum cpos have similar categorical properties to cpos and are therefore suitable for the construction of categorical models for quantum programming languages, which is illustrated with some examples. For this reason, quantum cpos may form the backbone of a future quantum domain theory.