The Structure of Sum-Over-Paths, its Consequences, and Completeness for Clifford

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 diag...

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Autor principal: Vilmart, Renaud
Formato: Online Artículo Texto
Lenguaje:English
Publicado: 2021
Materias:
Article
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7984103/
http://dx.doi.org/10.1007/978-3-030-71995-1_27
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author Vilmart, Renaud
author_facet Vilmart, Renaud
author_sort Vilmart, Renaud
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description 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.
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spelling pubmed-79841032021-03-23 The Structure of Sum-Over-Paths, its Consequences, and Completeness for Clifford Vilmart, Renaud Foundations of Software Science and Computation Structures Article 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. 2021-03-23 /pmc/articles/PMC7984103/ http://dx.doi.org/10.1007/978-3-030-71995-1_27 Text en © The Author(s) 2021 Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made. The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
spellingShingle Article
Vilmart, Renaud
The Structure of Sum-Over-Paths, its Consequences, and Completeness for Clifford
title The Structure of Sum-Over-Paths, its Consequences, and Completeness for Clifford
title_full The Structure of Sum-Over-Paths, its Consequences, and Completeness for Clifford
title_fullStr The Structure of Sum-Over-Paths, its Consequences, and Completeness for Clifford
title_full_unstemmed The Structure of Sum-Over-Paths, its Consequences, and Completeness for Clifford
title_short The Structure of Sum-Over-Paths, its Consequences, and Completeness for Clifford
title_sort structure of sum-over-paths, its consequences, and completeness for clifford
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7984103/
http://dx.doi.org/10.1007/978-3-030-71995-1_27
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