Abstract This paper unites two research lines. The first involves finding categorical models of quantum programming languages with recursion 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 with recursion. 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. Because of their noncommutative character, quantum cpos may form the backbone of a future quantum domain theory that provides structural methods for the denotational semantics of recursive quantum programming languages.

We consider a programming language that can manipulate both classical and quantum information. Our language is type-safe and designed for variational quantum programming, which is a hybrid classical-quantum computational paradigm. The classical subsystem of the language is the Probabilistic FixPoint Calculus (PFPC), which is a lambda calculus with mixed-variance recursive types, term recursion and probabilistic choice. The quantum subsystem is a first-order linear type system that can manipulate quantum information. The two subsystems are related by mixed classical/quantum terms that specify how classical probabilistic effects are induced by quantum measurements, and conversely, how classical (probabilistic) programs can influence the quantum dynamics. We also describe a sound and computationally adequate denotational semantics for the language. Classical probabilistic effects are interpreted using a recently-described commutative probabilistic monad on DCPO. Quantum effects and resources are interpreted in a category of von Neumann algebras that we show is enriched over (continuous) domains. This strong sense of enrichment allows us to develop novel semantic methods that we use to interpret the relationship between the quantum and classical probabilistic effects. By doing so we provide a very detailed denotational analysis that relates domain-theoretic models of classical probabilistic programming to models of quantum programming.