Y-Calculus: A language for real Matrices derived from the ZX-Calculus

The ZX-Calculus is a powerful diagrammatic language devoted to represent complex quantum evolutions. But the advantages of quantum computing still exist when working with rebits, and evolutions with real coefficients. Some models explicitly use rebits, but the ZX-Calculus can not handle these evolutions as it is. Hence, we define an alternative language solely dealing with real matrices, with a new set of rules. We show that three of its non-trivial rules are not derivable from the others and we prove that the language is complete for the π 2 -fragment. We define a generalisation of the Hadamard node, and exhibit two interpretations from and to the ZX-Calculus, showing the consistency between the two languages.