License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2016.76
URN: urn:nbn:de:0030-drops-65062
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6506/
Perdrix, Simon ;
Wang, Quanlong
Supplementarity is Necessary for Quantum Diagram Reasoning
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.
BibTeX - Entry
@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}
}
|
Keywords: |
|
quantum diagram reasoning, completeness, ZX-calculus, quantum computing, categorical quantum mechanics |
|
Collection: |
|
41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016) |
|
Issue Date: |
|
2016 |
|
Date of publication: |
|
19.08.2016 |
