On a recipe for quantum graphical languages

TL;DR

This paper classifies all two-dimensional quantum graphical calculi sharing a core structure called Z*-algebras, demonstrating they are variations of existing calculi like ZX, ZW, and ZH, and establishes the uniqueness of a related calculus.

Contribution

It provides a classification of Z*-algebras in 2D Hilbert spaces, showing they are essentially variations of known calculi and proving the uniqueness of a specific calculus.

Findings

All Z*-algebras in 2D are variations of ZX, ZW, and ZH calculi.

The calculus of linear relations is essentially unique.

The classification confirms no fundamentally new Z*-calculi will emerge beyond known variants.

Abstract

Different graphical calculi have been proposed to represent quantum computation. First the ZX- calculus [4], followed by the ZW-calculus [12] and then the ZH-calculus [1]. We can wonder if new Z*-calculi will continue to be proposed forever. This article answers negatively. All those language share a common core structure we call Z*-algebras. We classify Z*-algebras up to isomorphism in two dimensional Hilbert spaces and show that they are all variations of the aforementioned calculi. We do the same for linear relations and show that the calculus of [2] is essentially the unique one.