Minimality of the Stabilizer ZX Calculus

Abstract

The stabilizer fragment of the ZX calculus is amongst the most important fragments of the theory. Crucially, the stabilizer calculus can be described by a small collection of rewrites, most of which have been shown to be necessary by Backens--Perdrix--Wang (arXiv:1709.08903). However, two rules, describing the red/green compact-structure coincidence and the important bialgebra law, had not been shown to be necessary. We present a countermodel-style argument showing that both of these rules are individually necessary relative to the connectivity meta-rule, and hence establish that the rule set has no redundant rewrite rule.