The GHZ/W-calculus contains rational arithmetic

Graphical calculi provide an intuitive, compositional way to express and manipulate quantum states and processes. They also provide a bridge to automated techniques for reasoning and computation via graph rewriting. The power of these calculi stems from the fact that they subsume a wide range of symmetries in the structure of quantum operations such as 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 arithmetic, 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.