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Quantum Programs as Kleisli Maps

Bram Westerbaan

Published Dec 19, 2014• 1 citation
OpenAlex

Abstract

Abstract. In computer science the Kleisli category is often used to model computation with a twist such as nondeterminism or probabilism. Recently, Furber and Jacobs have shown in their study of quantum compu-

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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]
This paper uncovers the fundamental relationship between total and partial computation in the form of an equivalence of certain categories. This equivalence involves on the one hand effectuses, which are categories for total computation, introduced by Jacobs for the study of quantum/effect logic. On the other hand, it involves what we call FinPACs with effects; they are finitely partially additive categories equipped with effect algebra structures, serving as categories for partial computation. It turns out that the Kleisli category of the lift monad (−)+1 on an effectus is always a FinPAC with effects, and this construction gives rise to the equivalence. Additionally, state-and-effect triangles over FinPACs with effects are presented. 1
This paper uncovers the fundamental relationship between total and partial computation in the form of an equivalence of certain categories. This equivalence involves on the one hand effectuses, which are categories for total computation, introduced by Jacobs for the study of quantum/effect logic. On the other hand, it involves what we call FinPACs with effects; they are finitely partially additive categories equipped with effect algebra structures, serving as categories for partial computation. It turns out that the Kleisli category of the lift monad (-)+1 on an effectus is always a FinPAC with effects, and this construction gives rise to the equivalence. Additionally, state-and-effect triangles over FinPACs with effects are presented.