Abstract

The ZX-calculus is a powerful diagrammatic language for quantum mechanics and quantum information processing. We prove that its pi/4-fragment is not complete, in other words the ZX-calculus is not complete for the so called "Clifford+T quantum mechanics". The completeness of this fragment was one of the main open problems in categorical quantum mechanics, a programme initiated by Abramsky and Coecke. The ZX-calculus was known to be incomplete for quantum mechanics. On the other hand, its pi/2-fragment is known to be complete, i.e. the ZX-calculus is complete for the so called "stabilizer quantum mechanics". Deciding whether its pi/4-fragment is complete is a crucial step in the development of the ZX-calculus since this fragment is approximately universal for quantum mechanics, contrary to the pi/2-fragment.

To establish our incompleteness result, we consider a fairly simple property of quantum states called supplementarity. We show that supplementarity can be derived in the ZX-calculus if and only if the angles involved in this equation are multiples of pi/2. In particular, the impossibility to derive supplementarity for pi/4 implies the incompleteness of the ZX-calculus for Clifford+T quantum mechanics. As a consequence, we propose to add the supplementarity to the set of rules of the ZX-calculus.

We also show that if a ZX-diagram involves antiphase twins, they can be merged when the ZX-calculus is augmented with the supplementarity rule. Merging antiphase twins makes diagrammatic reasoning much easier and provides a purely graphical meaning to the supplementarity rule.

@InProceedings{perdrix_et_al:LIPIcs:2016:6506,
  author =	{Simon Perdrix and Quanlong Wang},
  title =	{{Supplementarity is Necessary for Quantum Diagram Reasoning}},
  booktitle =	{41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)},
  pages =	{76:1--76:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-016-3},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{58},
  editor =	{Piotr Faliszewski and Anca Muscholl and Rolf Niedermeier},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6506},
  URN =		{urn:nbn:de:0030-drops-65062},
  doi =		{10.4230/LIPIcs.MFCS.2016.76},
  annote =	{Keywords: quantum diagram reasoning, completeness, ZX-calculus, quantum computing, categorical quantum mechanics}
}