🤖 AI Summary
This paper addresses the lack of categorical semantics for quantum programming languages by introducing “quantum cpos”—the first systematic generalization of ω-complete partial orders (cpos) within a noncommutative mathematical framework. Methodologically, it internalizes classical domain-theoretic structures—such as directed completeness and limit existence—into the categories of von Neumann algebras and quantum relations via discrete quantization techniques. The primary contributions are: (i) the first categorical model unifying structural properties of classical cpos with intrinsic quantum features, thereby establishing a foundational skeleton for quantum domain theory; and (ii) the successful construction of semantic models for multiple quantum type systems, empirically validating their adequacy for modeling quantum program semantics. Collectively, this work achieves a systematic transfer of classical domain theory to quantum computation.
📝 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.