Complete ZX-calculi for the stabiliser fragment in odd prime dimensions

TL;DR

This paper develops complete ZX-calculi for the stabiliser fragment of quantum theory in odd prime dimensions, enabling comprehensive diagrammatic reasoning for these systems.

Contribution

It introduces a family of ZX-calculi for odd prime dimensions and proves their completeness, extending the qubit ZX-calculus features to higher dimensions.

Findings

Calculi recover many features of the qubit ZX-calculus in higher dimensions.

Proved the calculi are complete for stabiliser quantum operations.

Extended the calculus to mixed states with a discard construction.

Abstract

We introduce a family of ZX-calculi which axiomatise the stabiliser fragment of quantum theory in odd prime dimensions. These calculi recover many of the nice features of the qubit ZX-calculus which were lost in previous proposals for higher-dimensional systems. We then prove that these calculi are complete, i.e. provide a set of rewrite rules which can be used to prove any equality of stabiliser quantum operations. Adding a discard construction, we obtain a calculus complete for mixed state stabiliser quantum mechanics in odd prime dimensions, and this furthermore gives a complete axiomatisation for the related diagrammatic language for affine co-isotropic relations.