Semantics for variational Quantum programming

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 a very detailed denotational analysis that relates domain-theoretic models of classical probabilistic programming to models of quantum programming.

Quantum Expectation Transformers for Cost Analysis

“…In this equation, each probability weight associated to a final state is given by the reduction probability of to as determined by the operational semantics. This is precisely the statement of strong adequacy in the denotational semantics of probabilistic [12,17] and quantum programming languages [11,25].…”

“…As a first step towards achieving our main objective, we introduce a new domain-theoretic notion, called a cost structure (Section 2). It is based on Kegelspitzen [15], which are dcpo's (directed-complete partial orders) equipped with a suitable convex structure that may be used to reason about the semantics of probabilistic [12,26] and quantum programming languages [11]. A cost structure is then a pair (S, +) of a Kegelspitze S together with a cost addition operation + that allows us to model resource consumption in a coherent way.…”

A category of quantum posets

“…We note that in the unordered version of the model, which is founded on qSet instead of qCPO, we can interpret quantum channels as the Kleisli morphisms of a probabilistic monad on qSet [7, Corollary 25]. Consequently, quantum sets can be used to construct a categorical model for a hybrid quantum programming language whose classical subsystem is recursively typed and whose quantum subsystem is a first-order type system with recursive terms [21]. However, for a higher-order recursively-typed quantum programming language, unordered models do not seem to suffice.…”

Type-safe Quantum Programming in Idris

“…We now explain how the quantum program dynamics are represented in Qimaera in a type-safe way. We are (roughly) inspired by representing the notion of a quantum configuration as it appears in [32,29,22], which is in turn used to formally describe the operational semantics of quantum type systems. Qubits in Qimaera.…”