The GHZ/W-calculus contains rational arithmetic

TL;DR

This paper demonstrates that the GHZ/W-calculus, a graphical language for quantum systems, can encode standard rational arithmetic operations, highlighting its expressive power beyond traditional quantum representations.

Contribution

It shows that the GHZ/W-calculus can represent rational arithmetic, including operations like addition, multiplication, and inversion, using quantum states and gates.

Findings

GHZ state encodes multiplication

W state encodes addition

Pauli gates encode inversion operations

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.