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Addition and Differentiation of ZX-Diagrams

Authors Emmanuel Jeandel , Simon Perdrix , Margarita Veshchezerova

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Author Details

Emmanuel Jeandel
  • LORIA, CNRS, Université de Lorraine, Inria Mocqua, Nancy, France
Simon Perdrix
  • LORIA, CNRS, Université de Lorraine, Inria Mocqua, Nancy, France
Margarita Veshchezerova
  • LORIA, CNRS, Université de Lorraine, Inria Mocqua, Nancy, France
  • EDF R&D, France

Funding

This work was supported in part by the CIFRE EDF/Loria Quantum Computing for Combinatorial Optimisation, the French National Research Agency (ANR) under the research projects SoftQPro ANR-17-CE25-0009-02 and VanQuTe ANR-17-CE24-0035, by the DGE of the French Ministry of Industry under the research project PIA-GDN/QuantEx P163746-484124, by the PEPR integrated project EPiQ, by the STIC-AmSud project Qapla’ 21-STIC-10, and by the European projects NEASQC (funded from the European Union’s Horizon 2020 research and innovation programme grant agreement No 951821) and HPCQS (European High-Performance Computing Joint Undertaking under grant agreement No 101018180).

Acknowledgements

The authors want to thank Bob Coecke, Harny Wang, and Richie Yeung for their availability and the fruitful discussions in the last few days prior to the submission.

Cite As Get BibTex

Emmanuel Jeandel, Simon Perdrix, and Margarita Veshchezerova. Addition and Differentiation of ZX-Diagrams. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 13:1-13:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.FSCD.2022.13

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Axiomatic semantics
Keywords
  • ZX calculus
  • Addition of ZX diagrams
  • Diagrammatic differentiation

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