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

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 12650)

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.

Keywords

  • Categorical Quantum Mechanics
  • Dagger-Compact PROP
  • Sum-Over-Paths
  • Clifford Fragment
  • Normal Form
  • Rewriting
  • Discard Construction
  • Verification

This work was made during a Postdoc funded by the project PIA-GDN/Quantex. Proofs can be found at arXiv:2003.05678

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Vilmart, R. (2021). The Structure of Sum-Over-Paths, its Consequences, and Completeness for Clifford. In: Kiefer, S., Tasson, C. (eds) Foundations of Software Science and Computation Structures. FOSSACS 2021. Lecture Notes in Computer Science(), vol 12650. Springer, Cham. https://doi.org/10.1007/978-3-030-71995-1_27

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