Towards a Minimal Stabilizer ZX-calculus

TL;DR

This paper simplifies the stabilizer ZX-calculus by identifying a minimal set of necessary rules, clarifying which axioms are essential for its completeness in quantum reasoning.

Contribution

It demonstrates that most remaining rules are necessary for the calculus's completeness and explores the minimal categorical assumptions needed for the meta rule.

Findings

Most rules of the stabilizer ZX-calculus are necessary.

The bialgebra rule's necessity remains an open question.

A braided autonomous category suffices for the 'only connectivity matters' rule.

Abstract

The stabilizer ZX-calculus is a rigorous graphical language for reasoning about quantum mechanics. The language is sound and complete: one can transform a stabilizer ZX-diagram into another one using the graphical rewrite rules if and only if these two diagrams represent the same quantum evolution or quantum state. We previously showed that the stabilizer ZX-calculus can be simplified by reducing the number of rewrite rules, without losing the property of completeness [Backens, Perdrix & Wang, EPTCS 236:1--20, 2017]. Here, we show that most of the remaining rules of the language are indeed necessary. We do however leave as an open question the necessity of two rules. These include, surprisingly, the bialgebra rule, which is an axiomatisation of complementarity, the cornerstone of the ZX-calculus. Furthermore, we show that a weaker ambient category -- a braided autonomous category instead of the usual compact closed category -- is sufficient to recover the meta rule 'only connectivity matters', even without assuming any symmetries of the generators.