Applying quantitative semantics to higher-order quantum computing

Pagani, Michele; Selinger, Peter; Valiron, Benoît

doi:10.1145/2535838.2535879

Cited by 68 publications

(101 citation statements)

References 22 publications

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“…This makes a connection with other work on monoidal categories for quantum computation. The monad-based model seems to be closely related to the one proposed by [27]; it would be interesting to compare it with the model of [33]. One can also study the full subcategory of the Kleisli category that is spanned by objects of the form FΓ.…”

Section: Program Equationsmentioning

confidence: 95%

Algebraic Effects, Linearity, and Quantum Programming Languages

StatonSam

2015

SIGPLAN Not.

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We develop a new framework of algebraic theories with linear parameters, and use it to analyze the equational reasoning principles of quantum computing and quantum programming languages. We use the framework as follows:• we present a new elementary algebraic theory of quantum computation, built from unitary gates and measurement; • we provide a completeness theorem for the elementary algebraic theory by relating it with a model from operator algebra; • we extract an equational theory for a quantum programming language from the algebraic theory; • we compare quantum computation with other local notions of computation by investigating variations on the algebraic theory.

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Section: Program Equationsmentioning

confidence: 95%

Algebraic Effects, Linearity, and Quantum Programming Languages

StatonSam

2015

SIGPLAN Not.

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“…⊓ ⊔ If C is not normalised, then adequacy can be recovered simply by normalising: (⋄ • C ) (1) = tr(C)Halt(C), for any possible configuration C. The adequacy formulation of [17] and [4] is now a special case of our more general formulation. (M | · | · | 1) (1) = Halt(M | · | · | 1).…”

Section: ⊓ ⊔mentioning

confidence: 99%

“…Our work is the first to present a detailed semantic treatment of user-defined inductive datatypes for quantum programming. In [17] and [4], the authors show how to interpret a quantum lambda calculus extended with a datatype for lists, but their syntax does not support any other inductive datatypes. These languages are equipped with lambda abstractions, whereas our language has only support for procedures.…”

Section: Conclusion and Related Workmentioning

confidence: 99%

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Quantum Programming with Inductive Datatypes: Causality and Affine Type Theory

Péchoux

Perdrix

Rennela

et al. 2020

Lecture Notes in Computer Science

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Inductive datatypes in programming languages allow users to define useful data structures such as natural numbers, lists, trees, and others. In this paper we show how inductive datatypes may be added to the quantum programming language QPL. We construct a sound categorical model for the language and by doing so we provide the first detailed semantic treatment of user-defined inductive datatypes in quantum programming. We also show our denotational interpretation is invariant with respect to big-step reduction, thereby establishing another novel result for quantum programming. Compared to classical programming, this property is considerably more difficult to prove and we demonstrate its usefulness by showing how it immediately implies computational adequacy at all types. To further cement our results, our semantics is entirely based on a physically natural model of von Neumann algebras, which are mathematical structures used by physicists to study quantum mechanics.

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“…One way is to forbid the construction of the term λx (x ⊗ x) using a typing system inspired from linear logic [1,9], leading to logic-linear calculi [2,10,11,13,14]. Another is to consider all λ-terms expressing linear functions.…”

Section: Introductionmentioning

confidence: 99%

Typing Quantum Superpositions and Measurement

Díaz-Caro

Dowek

2017

Theory and Practice of Natural Computing

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Abstract. We propose a way to unify two approaches of non-cloning in quantum lambda-calculi. The first approach is to forbid duplicating variables, while the second is to consider all lambda-terms as algebraiclinear functions. We illustrate this idea by defining a quantum extension of first-order simply-typed lambda-calculus, where the type is linear on superposition, while allows cloning base vectors. In addition, we provide an interpretation of the calculus where superposed types are interpreted as vector spaces and non-superposed types as their basis.

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Applying quantitative semantics to higher-order quantum computing

Cited by 68 publications

References 22 publications

Algebraic Effects, Linearity, and Quantum Programming Languages

Algebraic Effects, Linearity, and Quantum Programming Languages

Quantum Programming with Inductive Datatypes: Causality and Affine Type Theory

Typing Quantum Superpositions and Measurement

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