Quantum Programs as Kleisli Maps

Abstract

Furber and Jacobs have shown in their study of quantum computation that the category of commutative C*-algebras and PU-maps (positive linear maps which preserve the unit) is isomorphic to the Kleisli category of a comonad on the category of commutative C*-algebras with MIU-maps (linear maps which preserve multiplication, involution and unit). [Furber and Jacobs, 2013] In this paper, we prove a non-commutative variant of this result: the category of C*-algebras and PU-maps is isomorphic to the Kleisli category of a comonad on the subcategory of MIU-maps. A variation on this result has been used to construct a model of Selinger and Valiron's quantum lambda calculus using von Neumann algebras. [Cho and Westerbaan, 2016]

In Proceedings QPL 2016, arXiv:1701.00242

References in corpus (2)

• From Kleisli Categories to Commutative C*-algebras: Probabilistic Gelfand Duality
• Von Neumann Algebras form a Model for the Quantum Lambda Calculus

Cited by in corpus (8)

• From Kleisli Categories to Commutative C*-algebras: Probabilistic Gelfand Duality
• Quantum Programming with Inductive Datatypes: Causality and Affine Type Theory
• The Category of Von Neumann Algebras
• Von Neumann Algebras form a Model for the Quantum Lambda Calculus
• Inverses, disintegrations, and Bayesian inversion in quantum Markov categories
• The Quantum Effect: A Recipe for QuantumPi
• Quantum Information Effects
• Duplicable von Neumann Algebras