The Simplified Stabilizer ZX-Calculus is Minimal

The stabilizer fragment of the ZX calculus is amongst the most important fragments of the theory. The closely related Clifford+T fragment is approximately universal (arXiv:1705.11151). Additionally, the stabilizer calculus can be described by a small collection of rewrites, most of which have been shown to be necessary (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 of Backens--Perdrix--Wang (arXiv:1709.08903), and hence establish that the rule set presented in arXiv:1709.08903 has no redundant rewrite rule.

Source: arXiv:2606.12383v1 - http://arxiv.org/abs/2606.12383v1 PDF: https://arxiv.org/pdf/2606.12383v1 Original Link: http://arxiv.org/abs/2606.12383v1