Full abstraction for the quantum lambda-calculus

Clairambault, Pierre; Visme, Marc de

doi:10.1145/3371131

Cited by 17 publications

(19 citation statements)

References 47 publications

(52 reference statements)

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“…We believe the results presented here are an important step towards bringing these two families together, aiming towards a unified landscape of quantitative denotational models of programming languages. We proved this for PCF, but there is no doubt that this extends to other languages or evaluation strategies -in fact, the first author and de Visme proved a similar collapse theorem for the (call-by-value) quantum λ-calculus [CdV20] (this relies on some of the constructions of this paper, first appearing in an unpublished technical report by the first author [Cla20]). Of course, much remains to be done: notably, we would like to understand better the links between thin concurrent games and generalized species of structure [FGHW08].…”

Section: Discussionmentioning

confidence: 76%

“…Absorption of Symmetries. As final contribution, we include a property which, though not used for the main results of this paper, was required for the quantum collapse of [CdV20]. As such, we believe it fits with the present development.…”

Section: 2mentioning

confidence: 67%

“…To our knowledge, the first quantitative collapse from games to relations is from probabilistic concurrent games to the relational model weighted by R + = R ∪ {+∞} in [CCPW18]. However, when working on an extension to a quantum language [CdV20], the first author discovered an error in [CCPW18]: the paper uses symmetry classes as witnesses, which -as we show here-is inadequate. The correct notion of witness and its validity w.r.t.…”

Section: Introductionmentioning

confidence: 96%

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The Quantitative Collapse of Concurrent Games with Symmetry

Clairambault,

Paquet

2021

Preprint

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We explore links between the thin concurrent games of Castellan, Clairambault and Winskel, and the weighted relational models of linear logic studied by Laird, Manzonetto, McCusker and Pagani. More precisely, we show that there is an interpretationpreserving "collapse" functor from the former to the latter. On objects, the functor defines for each game a set of possible execution states. Defining the action on morphisms is more subtle, and this is the main contribution of the paper.Given a strategy and an execution state, our functor needs to count the witnesses for this state within the strategy. Strategies in thin concurrent games describe non-linear behaviour explicitly, so in general each witness exists in countably many symmetric copies. The challenge is to define the right notion of witnesses, factoring out this infinity while matching the weighted relational model. Understanding how witnesses compose is particularly subtle and requires a delve into the combinatorics of witnesses and their symmetries.In its basic form, this functor connects thin concurrent games and a relational model weighted by N ∪ {+∞}. We will additionally consider a generalised setting where both models are weighted by elements of an arbitrary continuous semiring; this covers the probabilistic case, among others. Witnesses now additionally carry a value from the semiring, and our interpretation-preserving collapse functor extends to this setting.

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Section: Discussionmentioning

confidence: 76%

Section: 2mentioning

confidence: 67%

Section: Introductionmentioning

confidence: 96%

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The Quantitative Collapse of Concurrent Games with Symmetry

Clairambault,

Paquet

2021

Preprint

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“…Our approach also differs from that taken in games semantics, which has been highly successful in constructing fully abstract models for a variety of languages (e.g. Abramsky et al [1998Abramsky et al [ , 2000; Clairambault and de Visme [2020]; Hyland and Ong [2000]). In such models the notion of morphism is changed to include more intensional information, and one obtains a fully abstract model by identifying suitable morphisms.…”

Section: Related Workmentioning

confidence: 99%

Fully abstract models for effectful λ-calculi via category-theoretic logical relations

Kammar

Katsumata

Saville

2022

Proc. ACM Program. Lang.

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We present a construction which, under suitable assumptions, takes a model of Moggi’s computational λ-calculus with sum types, effect operations and primitives, and yields a model that is adequate and fully abstract. The construction, which uses the theory of fibrations, categorical glueing, ⊤⊤-lifting, and ⊤⊤-closure, takes inspiration from O’Hearn & Riecke’s fully abstract model for PCF. Our construction can be applied in the category of sets and functions, as well as the category of diffeological spaces and smooth maps and the category of quasi-Borel spaces, which have been studied as semantics for differentiable and probabilistic programming.

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“…Semantics for Variational Quantum Programming 26:29 8.2 Quantum Lambda Calculi Other related work includes adequate [Tsukada et al 2018] and even fully-abstract semantics for the quantum lambda calculus [Clairambault and de Visme 2020;Clairambault et al 2019;Pagani et al 2014]. These models are inspired by quantitative models of linear logic and we will collectively refer to them as the quantitative models.…”

mentioning

confidence: 99%

Semantics for variational Quantum programming

Jia

Kornell

Lindenhovius³

et al. 2022

Proc. ACM Program. Lang.

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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.

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Full abstraction for the quantum lambda-calculus

Cited by 17 publications

References 47 publications

The Quantitative Collapse of Concurrent Games with Symmetry

The Quantitative Collapse of Concurrent Games with Symmetry

Fully abstract models for effectful λ-calculi via category-theoretic logical relations

Semantics for variational Quantum programming

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