Realizability in the Unitary Sphere

@article{DazCaro2019RealizabilityIT,
title={Realizability in the Unitary Sphere},
author={Alejandro D{\'i}az-Caro and M. Guillermo and Alexandre Miquel and B. Valiron},
journal={2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)},
year={2019},
pages={1-13}
}
• Alejandro Díaz-Caro, +1 author B. Valiron
• Published 2019
• Computer Science, Mathematics
• 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
In this paper we present a semantics for a linear algebraic lambda-calculus based on realizability. This semantics characterizes a notion of unitarity in the system, answering a long standing issue. We derive from the semantics a set of typing rules for a simply-typed linear algebraic lambda-calculus, and show how it extends both to classical and quantum lambda-calculi.
8 Citations

Tables and Topics from this paper

A fully abstract model for quantum controlled lambda calculus
• Mathematics
• 2018
We give a fully-abstract, concrete, categorical model for Lambda-S. Lambda-S is an extension to first-order lambda calculus unifying two approaches of non-cloning in quantum lambda-calculi: to forbidExpand
The Vectorial Lambda Calculus Revisited
• Computer Science, Mathematics
• ArXiv
• 2020
It is proved that the revised Vectorial Lambda Calculus supports the standard version of the Subject Reduction property, and also introduces the concept of weight of types and terms, and proves a relation between the weight of terms and of its types. Expand
A Categorical Construction for the Computational Definition of Vector Spaces
• Mathematics, Computer Science
• Appl. Categorical Struct.
• 2020
An abstract categorical semantics of Lambda- S is given, showing that S can be interpreted as the composition of two functors in an adjunction relation between a Cartesian category and an additive symmetric monoidal category. Expand
PhD Proposal
Quantum Computing is a new and emerging computational paradigm whose main idea is to use quantum mechanical phenomena, such as entanglement and superposition, in order to perform computation. AExpand
A New Connective in Natural Deduction, and its Application to Quantum Computing
• Computer Science
• ICTAC
• 2021
This work introduces a propositional logic with a non harmonious connective sup, proves cut elimination for this logic, and shows that its proof language forms the core of a quantum programming language. Expand
Extensional proofs in a propositional logic modulo isomorphisms
• Computer Science, Mathematics
• ArXiv
• 2020
It is shown here that adding $\eta$-expansion rules to System I permits to drop this restriction, and yields a strongly normalising calculus with enjoying the full introduction property. Expand
Polymorphic System I
• Computer Science
• IFL
• 2020
This work proposes an extension of System I to polymorphic types, adding the corresponding isomorphisms, and provides non-standard proofs of subject reduction and strong normalisation, extending those of system I. Expand
Quantum Control in the Unitary Sphere: Lambda-S1 and its Categorical Model
• Computer Science, Mathematics
• ArXiv
• 2020
This paper introduces a valid subset of typing rules, defining an expressive enough quantum calculus, and proposes a categorical semantics for it, which consists of an adjunction between the category of semi-vector spaces of value distributions and the categories of sets of value distribution. Expand

References

SHOWING 1-10 OF 31 REFERENCES
A Lambda Calculus for Quantum Computation with Classical Control
• Computer Science
• TLCA
• 2005
The main results of this paper are the safety properties of the language and the development of a type inference algorithm. Expand
Applying quantitative semantics to higher-order quantum computing
• Computer Science, Physics
• POPL 2014
• 2013
This paper proposes a denotational semantics for a quantum lambda calculus with recursion and an infinite data type, using constructions from quantitative semantics of linear logic. Expand
A typed, algebraic, computational lambda-calculus†
• B. Valiron
• Computer Science
• Mathematical Structures in Computer Science
• 2013
This paper develops a categorical analysis of a general simply typed lambda-calculus endowed with the structure of a module and develops various concrete models for both the case without fixpoints and for the case with them. Expand
On a Fully Abstract Model for a Quantum Linear Functional Language: (Extended Abstract)
• Computer Science
• Electron. Notes Theor. Comput. Sci.
• 2008
This paper studies the linear fragment of the programing language for quantum computation with classical control described in, and sketches the language, and describes a fully abstract denotational semantics based on completely positive maps. Expand
The algebraic lambda calculus
• L. Vaux
• Computer Science, Mathematics
• Mathematical Structures in Computer Science
• 2009
An extension of the pure lambda calculus is introduced by endowing the set of terms with the structure of a vector space, or, more generally, of a module, over a fixed set of scalars, and it is proved it is confluent. Expand
Presheaf Models of Quantum Computation: An Outline
• Mathematics, Computer Science
• Computation, Logic, Games, and Quantum Foundations
• 2013
A concrete denotational semantics of Selinger and Valiron’s quantum lambda calculus is constructed by considering presheaves over appropriate base categories arising from first-order quantum computation. Expand
The differential lambda-calculus
• Mathematics, Computer Science
• Theor. Comput. Sci.
• 2003
We present an extension of the lambda-calculus with differential constructions. We state and prove some basic results (confluence, strong normalization in the typed case), and also a theorem relatingExpand
Lineal: A linear-algebraic Lambda-calculus
• Computer Science, Physics
• Log. Methods Comput. Sci.
• 2017
A minimal language combining higher-order computation and linear algebra is introduced, which extends the Lambda-calculus with the possibility to make arbitrary linear combinations of terms alpha.t + beta.u.t and the confluence of the entire calculus is proved. Expand
Linear-algebraic λ-calculus: higher-order, encodings, and confluence
We introduce a minimal language combining higher-order computation and linear algebra. This language extends the λ-calculus with the possibility to make arbitrary linear combinations of terms α.t+Expand
Linearity in the Non-deterministic Call-by-Value Setting
• Mathematics, Computer Science
• WoLLIC
• 2012
A fine-grained type system is defined, capturing the right linearity present in such formalisms as the call-by-value lambda calculus, which corresponds to the additive fragment of the linear-algebraic lambda-calculus. Expand