Towards a Minimal Stabilizer ZX-calculus

摘要

The stabilizer ZX-calculus is a rigorous graphical language for reasoningabout quantum mechanics. The language is sound and complete: one can transforma stabilizer ZX-diagram into another one using the graphical rewrite rules ifand only if these two diagrams represent the same quantum evolution or quantumstate. We previously showed that the stabilizer ZX-calculus can be simplifiedby reducing the number of rewrite rules, without losing the property ofcompleteness [Backens, PerdrixandWang, EPTCS 236:1--20, 2017]. Here, we showthat most of the remaining rules of the language are indeed necessary. We dohowever leave as an open question the necessity of two rules. These include,surprisingly, the bialgebra rule, which is an axiomatisation ofcomplementarity, the cornerstone of the ZX-calculus. Furthermore, we show thata weaker ambient category -- a braided autonomous category instead of the usualcompact closed category -- is sufficient to recover the meta rule 'onlyconnectivity matters', even without assuming any symmetries of the generators.