Quantum Programs as Kleisli Maps

TL;DR

This paper extends a known categorical equivalence from commutative to non-commutative C*-algebras, providing a foundation for quantum lambda calculus models using von Neumann algebras.

Contribution

It proves a non-commutative variant of a categorical isomorphism involving C*-algebras and PU-maps, broadening the mathematical framework for quantum computation.

Findings

Established isomorphism between C*-algebras and Kleisli categories in the non-commutative setting

Extended categorical models to von Neumann algebras for quantum lambda calculus

Provided theoretical foundations for quantum programming language semantics

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]