ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T quantum mechanics

TL;DR

This paper introduces cyclotomic supplementarity as a new rule in the ZX-Calculus, demonstrating its necessity for completeness and showing that the existing calculus is incomplete for Clifford+T quantum mechanics.

Contribution

It presents a generalized supplementarity rule, proves its limitations, and establishes the incompleteness of the ZX-Calculus for Clifford+T quantum mechanics, proposing an extended axiomatisation.

Findings

Cyclotomic supplementarity cannot be derived for odd prime n.

An additional simple axiom is necessary for completeness.

The ZX-Calculus is incomplete for Clifford+T quantum mechanics.

Abstract

The ZX-Calculus is a powerful graphical language for quantum mechanics and quantum information processing. The completeness of the language -- i.e. the ability to derive any true equation -- is a crucial question. In the quest of a complete ZX-calculus, supplementarity has been recently proved to be necessary for quantum diagram reasoning (MFCS 2016). Roughly speaking, supplementarity consists in merging two subdiagrams when they are parameterized by antipodal angles. We introduce a generalised supplementarity -- called cyclotomic supplementarity -- which consists in merging n subdiagrams at once, when the n angles divide the circle into equal parts. We show that when n is an odd prime number, the cyclotomic supplementarity cannot be derived, leading to a countable family of new axioms for diagrammatic quantum reasoning.We exhibit another new simple axiom that cannot be derived from the existing rules of the ZX-Calculus, implying in particular the incompleteness of the language for the so-called Clifford+T quantum mechanics. We end up with a new axiomatisation of an extended ZX-Calculus, including an axiom schema for the cyclotomic supplementarity.