Y-Calculus: A Language for Real Matrices Derived from the ZX-Calculus

TL;DR

Y-Calculus is a new diagrammatic language for real matrices that extends ZX-Calculus, with proven axioms, completeness results, and interpretations linking the two formalisms, enabling extraction of real and imaginary parts of diagrams.

Contribution

It introduces Y-Calculus, a ZX-like language for real matrices, with axioms, completeness, and interpretations connecting it to ZX-Calculus, advancing diagrammatic reasoning in quantum computation.

Findings

Established non-trivial axioms for Y-Calculus.

Proved completeness of certain language restrictions.

Developed interpretations between Y-Calculus and ZX-Calculus.

Abstract

We introduce a ZX-like diagrammatic language devoted to manipulating real matrices - and rebits -, with its own set of axioms. We prove the necessity of some non trivial axioms of these. We show that some restriction of the language is complete. We exhibit two interpretations to and from the ZX-Calculus, thus showing the consistency between the two languages. Finally, we derive from our work a way to extract the real or imaginary part of a ZX-diagram, and prove that a restriction of our language is complete if the equivalent restriction of the ZX-calculus is complete.