A Diagrammatic Axiomatisation of Fermionic Quantum Circuits

Authors Amar Hadzihasanovic, Giovanni de Felice, Kang Feng Ng

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Author Details

Amar Hadzihasanovic
  • RIMS, Kyoto University, Japan
Giovanni de Felice
  • Department of Computer Science, University of Oxford, United Kingdom
Kang Feng Ng
  • Department of Computer Science, University of Oxford, United Kingdom

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Amar Hadzihasanovic, Giovanni de Felice, and Kang Feng Ng. A Diagrammatic Axiomatisation of Fermionic Quantum Circuits. In 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 108, pp. 17:1-17:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


We introduce the fermionic ZW calculus, a string-diagrammatic language for fermionic quantum computing (FQC). After defining a fermionic circuit model, we present the basic components of the calculus, together with their interpretation, and show how the main physical gates of interest in FQC can be represented in the language. We then list our axioms, and derive some additional equations. We prove that the axioms provide a complete equational axiomatisation of the monoidal category whose objects are quantum systems of finitely many local fermionic modes, with operations that preserve or reverse the parity (number of particles mod 2) of states, and the tensor product, corresponding to the composition of two systems, as monoidal product. We achieve this through a procedure that rewrites any diagram in a normal form. We conclude by showing, as an example, how the statistics of a fermionic Mach-Zehnder interferometer can be calculated in the diagrammatic language.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Fermionic Quantum Computing
  • String Diagrams
  • Categorical Quantum Mechanics


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