Categories of quantum cpos
For researchers in quantum programming languages and domain theory, this work provides a foundational structure (quantum cpos) that could enable a quantum domain theory, though it is an initial proposal with no concrete performance metrics.
This paper introduces quantum cpos, a noncommutative generalization of ω-complete partial orders, using discrete quantization in a category of von Neumann algebras and quantum relations. It demonstrates that quantum cpos have categorical properties similar to cpos, making them suitable for constructing categorical models of quantum programming languages.
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 $ω$-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.