Making the stabilizer ZX-calculus complete for scalars

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

12 Citations (Scopus)


The ZX-calculus is a graphical language for quantum processes with built-in rewrite rules. The rewrite rules allow equalities to be derived entirely graphically, leading to the question of completeness: can any equality that is derivable using matrices also be derived graphically? The ZX-calculus is known to be complete for scalar-free pure qubit stabilizer quantum mechanics, meaning any equality between two pure stabilizer operators that is true up to a non-zero scalar factor
can be derived using the graphical rewrite rules. Here, we replace those scalar-free rewrite rules with correctly scaled ones and show that, by adding one new diagram element and a new rewrite rule, the calculus can be made complete for pure qubit stabilizer quantum mechanics with scalars. This completeness property allows amplitudes and probabilities to be calculated entirely graphically. We also explicitly consider stabilizer zero diagrams, i.e. diagrams that represent a zero matrix, and show that two new rewrite rules suffice to make the calculus complete for those.
Original languageEnglish
Title of host publicationProceedings of the 12th International Workshop on Quantum Physics and Logic (QPL 2015)
EditorsChris Heunen, Peter Selinger, Jamie Vicary
PublisherOpen Publishing Association
Publication statusPublished - 4 Nov 2015
Event12th International Workshop on Quantum Physics and Logic (QPL 2015) - Oxford, United Kingdom
Duration: 15 Jul 201517 Jul 2015

Publication series

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


Conference12th International Workshop on Quantum Physics and Logic (QPL 2015)
Country/TerritoryUnited Kingdom


Dive into the research topics of 'Making the stabilizer ZX-calculus complete for scalars'. Together they form a unique fingerprint.

Cite this