Semantics for variational Quantum programming

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