Emmanuel Jeandel ; Simon Perdrix ; Margarita Veshchezerova - Addition and Differentiation of ZX-diagrams

lmcs:11049 - Logical Methods in Computer Science, May 20, 2024, Volume 20, Issue 2 - https://doi.org/10.46298/lmcs-20(2:10)2024
Addition and Differentiation of ZX-diagramsArticle

Authors: Emmanuel Jeandel ; Simon Perdrix ; Margarita Veshchezerova

    The ZX-calculus is a powerful framework for reasoning in quantum computing. It provides in particular a compact representation of matrices of interests. A peculiar property of the ZX-calculus is the absence of a formal sum allowing the linear combinations of arbitrary ZX-diagrams. The universality of the formalism guarantees however that for any two ZX-diagrams, the sum of their interpretations can be represented by a ZX-diagram. We introduce a general, inductive definition of the addition of ZX-diagrams, relying on the construction of controlled diagrams. Based on this addition technique, we provide an inductive differentiation of ZX-diagrams. Indeed, given a ZX-diagram with variables in the description of its angles, one can differentiate the diagram according to one of these variables. Differentiation is ubiquitous in quantum mechanics and quantum computing (e.g. for solving optimization problems). Technically, differentiation of ZX-diagrams is strongly related to summation as witnessed by the product rules. We also introduce an alternative, non inductive, differentiation technique rather based on the isolation of the variables. Finally, we apply our results to deduce a diagram for an Ising Hamiltonian.

    Volume: Volume 20, Issue 2
    Published on: May 20, 2024
    Accepted on: January 15, 2024
    Submitted on: March 9, 2023
    Keywords: Quantum Physics

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