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DOI: 10.4230/LIPIcs.MFCS.2017.11
URN: urn:nbn:de:0030-drops-81173
URL: http://drops.dagstuhl.de/opus/volltexte/2017/8117/
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Jeandel, Emmanuel ; Perdrix, Simon ; Vilmart, Renaud ; Wang, Quanlong

ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T Quantum Mechanics

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LIPIcs-MFCS-2017-11.pdf (0.6 MB)


Abstract

The ZX-Calculus is a powerful graphical language for quantum mechanics and quantum information processing. The completeness of the language - i.e. the ability to derive any true equation - is a crucial question. In the quest of a complete ZX-calculus, supplementarity has been recently proved to be necessary for quantum diagram reasoning (MFCS 2016). Roughly speaking, supplementarity consists in merging two subdiagrams when they are parameterized by antipodal angles. We introduce a generalised supplementarity - called cyclotomic supplementarity - which consists in merging n subdiagrams at once, when the n angles divide the circle into equal parts. We show that when n is an odd prime number, the cyclotomic supplementarity cannot be derived, leading to a countable family of new axioms for diagrammatic quantum reasoning. We exhibit another new simple axiom that cannot be derived from the existing rules of the ZX-Calculus, implying in particular the incompleteness of the language for the so-called Clifford+T quantum mechanics. We end up with a new axiomatisation of an extended ZX-Calculus, including an axiom schema for the cyclotomic supplementarity.

BibTeX - Entry

@InProceedings{jeandel_et_al:LIPIcs:2017:8117,
  author =	{Emmanuel Jeandel and Simon Perdrix and Renaud Vilmart and Quanlong Wang},
  title =	{{ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T Quantum Mechanics}},
  booktitle =	{42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)},
  pages =	{11:1--11:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-046-0},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{83},
  editor =	{Kim G. Larsen and Hans L. Bodlaender and Jean-Francois Raskin},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/8117},
  URN =		{urn:nbn:de:0030-drops-81173},
  doi =		{10.4230/LIPIcs.MFCS.2017.11},
  annote =	{Keywords: Categorical Quantum Mechanincs, ZX-Calculus, Completeness, Cyclotomic Supplmentarity, Clifford+T}
}

Keywords: Categorical Quantum Mechanincs, ZX-Calculus, Completeness, Cyclotomic Supplmentarity, Clifford+T
Seminar: 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)
Issue Date: 2017
Date of publication: 22.11.2017


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