Semantics for Variational Quantum Programming

Speaker: Vladimir Zamdzhiev, Inria
Tuesday 15 March 2022, 11:00, (online)

Abstract: In this talk, I will first introduce myself by briefly summarising my main research activities: (1) formal methods for diagrammatic reasoning about quantum information processing (e.g. higher-order rewriting of ZX-calculus diagrams); and (2) design and analysis of quantum programming languages. I will also explain how this fits perfectly within the research activities of the QuaCS team.

After that I will talk about some recent work on variational quantum programming that will appear in POPL 2022. The technical summary of this talk is below.

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.