Differential categories

Summary. The introduction of linear logic and its associated proof theory has revolutionized many semantical investigations, for example, the search for fullyabstract models of PCF and the analysis of optimal reduction strategies for lambda calculi. In the present paper we show how proof nets, a graph-theoretic syntax for linear logic proofs, can be interpreted as operators in a simple calculus. This calculus was inspired by Feynman diagrams in quantum field theory and is accordingly called the φ-calculus. The ingredients are formal integrals, formal power series, a derivative-like construct and analogues of the Dirac delta function.Many of the manipulations of proof nets can be understood as manipulations of formulas reminiscent of a beginning calculus course. In particular, the "box" construct behaves like an exponential and the nesting of boxes phenomenon is the analogue of an exponentiated derivative formula. We show that the equations for the multiplicative-exponential fragment of linear logic hold.

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“…The possible relationships to the present work are striking. Ehrhard and Regnier's work was subsequently categorified in [BCS06].…”

Section: Discussionmentioning

confidence: 99%

Proof Nets as Formal Feynman Diagrams

Blute

Panangaden

2010

New Structures for Physics

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“…A completely different interpretation of the categorical structures described in this paper is as a model for the differential calculus of polynomials in one or more variables (see [3], and also [4] developed in parallel to this work). In this viewpoint, the raising operator performs multiplication by a homogeneous polynomial of degree 1, and the lowering operator performs differentiation.…”

Section: Discussionmentioning

confidence: 99%

“…We also see that the adjunction R Q gives F the structure of a comonad. 3 A related approach, developed in parallel to this work by another author [4] is to consider the comonad (F, , RηQ) as primary rather than the adjunction. This is a more general framework, but one in which the counit morphisms Rη (A,g,u)× will not be available for all commutative comonoids (A, g, u) × .…”

Section: The Categorical Frameworkmentioning

confidence: 99%

A Categorical Framework for the Quantum Harmonic Oscillator

Vicary

2008

Int J Theor Phys

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This paper describes how the structure of the state space of the quantum harmonic oscillator can be described by an adjunction of categories, that encodes the raising and lowering operators into a commutative comonoid. The formulation is an entirely general one in which Hilbert spaces play no special role.Generalised coherent states arise through the hom-set isomorphisms defining the adjunction, and we prove that they are eigenstates of the lowering operators. Generalised exponentials also emerge naturally in this setting, and we demonstrate that coherent states are produced by the exponential of a raising morphism acting on the zero-particle state. Finally, we examine all of these constructions in a suitable category of Hilbert spaces, and find that they reproduce the conventional mathematical structures.

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“…If the two logical systems do not match precisely, they match enough to be able to adapt models of linear logic to the subset of quantum computation described by the quantum lambda-calculus. Specically, many models of linear logic are based on functional analysis and theory of operator spaces, and use the algebraic linearity to reflect the logical linearity 10,5,12,11) . The spaces used for the models are rich enough to be able to generate the modalities "!"…”

Section: 1] Completely Positive Mapsmentioning

confidence: 99%

Quantum Computation: From a Programmer’s Perspective

Valiron

2013

New Gener. Comput.

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This paper is the second part of a series of two articles on quantum computation. If the first part was mostly concerned with the mathematical formalism, here we turn to the programmer's perspective. We analyze the various existing models of quantum computation and the problem of the stability of quantum information. We discuss the needs and challenges for the design of a scalable quantum programming language. We then present two interesting approaches and examine their strengths and weaknesses. Finally, we take a step back, and review the state of the research on the semantics of quantum computation, and how this can help in achieving some of the goals.Keywords Quantum Computation, Models of Computation, Quantum Programming Language, Semantics. §1 Introduction This paper is the second part of a two-articles series. The first part 30) was concerned with a general introduction to quantum computation; in this article, we discus quantum computation specifically from a programmer's perspective.Supported by the Intelligence Advanced Research Projects Activity (IARPA) via Department of Interior National Business Center contract number D11PC20168. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. Disclaimer: The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of IARPA, DoI/NBC, or the U.S. Government. Benoît ValironIn particular, we assume that the quantum device is already scalable and that we already know which algorithm we want to implement. The questions we want to be able to answer are how to actually write the algorithm and how to make sure that it is correct, and that it will run as expected.The paper is organized as follows. First we discuss the distinction between the abstract models of computation in which algorithms are usually defined; then we focus on the aspect that is not addressed in these scheme, that is, the problem of error detection and correction. We then turn to the question of the programming of quantum devices, from the perspective of the discussed computational models. We first discuss the need for an actual quantum programming language: why we need one, what can we expect from it. Then we develop two attempts at building a language, and discuss their strengths, their weaknesses and what lessons can be kept for future work.To conclude the paper, we analyze what have been done in the semantics of quantum computation: how they relate to models of computation, and what benefits these studies can bring. §2 Ideal versus realistic models of computationIn this section, we turn to the question of setting up a robust setting for performing quantum computation. There are two sides to this coin: the first is to define a logical layer, resilient to decoherence, and the other is the definition of a paradigm for quantum computation, with ideal quantum bits. Models ...

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Differential categories

Cited by 96 publications

References 24 publications

Proof Nets as Formal Feynman Diagrams

Proof Nets as Formal Feynman Diagrams

A Categorical Framework for the Quantum Harmonic Oscillator

Quantum Computation: From a Programmer’s Perspective

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