# The GHZ/W−calculus contains rational arithmetic

*Bob Coecke‚ Aleks Kissinger‚ Alex Merry and Shibdas Roy*

### Abstract

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.

Book Title

Electronic Proceedings in Theoretical Computer Science

Pages

34−48

Volume

52

Year

2011