Measurements in proof nets as higher-order quantum circuits

Abstract

We build on the series of work by Dal Lago and coauthors and identify proof nets (of linear logic) as higher-order quantum circuits. By accommodating quantum measurement using additive slices, we obtain a comprehensive framework for programming and interpreting quantum computation. Specifically, we introduce a quantum lambda calculus MLLqm and define its geometry of interaction (GoI) semantics - in the style of token machines - via the translation of terms into proof nets. Its soundness, i.e. invariance under reduction of proof nets, is established. The calculus MLLqm attains a pleasant balance between expressivity (it is higher-order and accommodates all quantum operations) and concreteness of models (given as token machines, i.e. in the form of automata). © 2014 Springer-Verlag.