Semantics for Variational Quantum Programming

TL;DR

This paper introduces a type-safe language for variational quantum programming that integrates classical probabilistic and quantum effects, providing a denotational semantics linking classical and quantum models.

Contribution

It presents the first denotational semantics connecting classical probabilistic programming models with quantum programming models in a unified framework.

Findings

Develops a hybrid classical-quantum programming language with sound semantics.

Uses a probabilistic monad and von Neumann algebras for semantic interpretation.

Provides a novel semantic method relating classical and quantum effects.

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.