The GHZ/W-calculus contains rational arithmetic

Bob Coecke
Aleks Kissinger
Alex Merry
Shibdas Roy

Graphical calculi for representing interacting quantum systems serve a number of purposes: compositionally, intuitive graphical reasoning, and a logical underpinning for automation. The power of these calculi stems from the fact that they embody generalized symmetries of the structure of quantum operations, which, for example, stretch well beyond the Choi-Jamiolkowski isomorphism. One such calculus takes the GHZ and W states as its basic generators. Here we show that this language allows one to encode standard rational calculus, with the GHZ state as multiplication, the W state as addition, the Pauli X gate as multiplicative inversion, and the Pauli Z gate as additive inversion.

Invited Presentation in Farid Ablayev, Bob Coecke and Alexander Vasiliev: Proceedings CSR 2010 Workshop on High Productivity Computations (HPC 2010), Kazan, Russia, June 21-22, 2010, Electronic Proceedings in Theoretical Computer Science 52, pp. 34–48.
Published: 9th March 2011.

ArXived at: http://dx.doi.org/10.4204/EPTCS.52.4 bibtex PDF
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