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Minimality of the Pure Qubit ZX Calculus
Harry K. Stoltz, Renaud Vilmart·August 14, 2026
Quantum Physics
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Abstract
The ZX calculus is a graphical language for reasoning about quantum processes. In this paper, we develop a minimal pure-qubit ZX calculus based on the work of Vilmart [arXiv:1812.09114], Backens, Perdrix, and Wang [arXiv:1709.08903], and Stoltz [arXiv:2606.12383]. This resolves a problem that has remained open for nearly a decade, since completeness was first proved. Specifically, we show that $(I_r)$ is derivable and establish the necessity of $(B)$ and $(I_g)$, yielding two complete and minimal rulesets.