ZH: a complete graphical calculus for quantum computations involving classical non-linearity

Research output: Chapter in Book/Report/Conference proceedingConference contribution


Colleges, School and Institutes

External organisations

  • Radboud University
  • University of Oxford


We present a new graphical calculus that is sound and complete for a universal family of quantum circuits, which can be seen as the natural string-diagrammatic extension of the approximately (real-valued) universal family of Hadamard+CCZ circuits. The diagrammatic language is generated by two kinds of nodes: the so-called 'spider' associated with the computational basis, as well as a new arity-N generalisation of the Hadamard gate, which satisfies a variation of the spider fusion law. Unlike previous graphical calculi, this admits compact encodings of non-linear classical functions. For example, the AND gate can be depicted as a diagram of just 2 generators, compared to ~25 in the ZX-calculus. Consequently, N-controlled gates, hypergraph states, Hadamard+Toffoli circuits, and diagonal circuits at arbitrary levels of the Clifford hierarchy also enjoy encodings with low constant overhead. This suggests that this calculus will be significantly more convenient for reasoning about the interplay between classical non-linear behaviour (e.g. in an oracle) and purely quantum operations. After presenting the calculus, we will prove it is sound and complete for universal quantum computation by demonstrating the reduction of any diagram to an easily describable normal form.


Original languageEnglish
Title of host publicationProceedings of the 15th International Conference on Quantum Physics and Logic (QPL 2018)
EditorsPeter Selinger, Chiribella Giulio
Publication statusPublished - 29 Jan 2019
Event15th International Conference on Quantum Physics and Logic (QPL 2018) - Dalhousie University, Halifax, Canada
Duration: 3 Jun 20187 Jun 2018

Publication series

NameElectronic Proceedings in Theoretical Computer Science
PublisherOpen Publishing Association
ISSN (Electronic)2075-2180


Conference15th International Conference on Quantum Physics and Logic (QPL 2018)

ASJC Scopus subject areas