The Structure of Sum-Over-Paths, its Consequences, and Completeness for Clifford
Renaud Vilmart
Abstract
Abstract We show that the formalism of “Sum-Over-Path” (SOP), used for symbolically representing linear maps or quantum operators, together with a proper rewrite system, has the structure of a dagger-compact PROP. Several consequences arise from this observation: – Morphisms of SOP are very close to the diagrams of the graphical calculus called ZH-Calculus, so we give a system of interpretation between the two – A construction, called the discard construction, can be applied to enrich the formalism so that, in particular, it can represent the quantum measurement. We also enrich the rewrite system so as to get the completeness of the Clifford fragments of both the initial formalism and its enriched version.
Topics & Concepts
MorphismFormalism (music)Computer scienceAlgebra over a fieldCompleteness (order theory)Calculus (dental)Theoretical computer scienceDiscrete mathematicsMathematicsPure mathematicsDentistryVisual artsMathematical analysisMedicineArtMusicalQuantum Mechanics and ApplicationsQuantum Computing Algorithms and ArchitectureLogic, programming, and type systems