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Semantics for Variational Quantum Programming

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Abstract

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 the first denotational analysis that relates models of classical probabilistic programming to models of quantum programming.

Dates and versions

hal-03519235 , version 1 (10-01-2022)

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Xiaodong Jia, Andre Kornell, Bert Lindenhovius, Michael Mislove, Vladimir Zamdzhiev. Semantics for Variational Quantum Programming. POPL 2022 - 49th ACM SIGPLAN Symposium on Principles of Programming Languages, Jan 2022, Philadelphia, United States. ⟨10.1145/3498687⟩. ⟨hal-03519235⟩
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