PreprintCC BY 4.0arXiv
Minimality of the Pure Qubit ZX Calculus
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 (Ir) is derivable and establish the necessity of (B) and (Ig), yielding two complete and minimal rulesets.
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BibTeX
@misc{stoltz2026minimality,
title = {{Minimality of the Pure Qubit ZX Calculus}},
author = {Harry K. Stoltz and Renaud Vilmart},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.14872},
eprint = {2608.14872},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.14872},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Minimality of the Pure Qubit ZX Calculus
AU - Harry K. Stoltz
AU - Renaud Vilmart
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2608.14872
UR - https://arxiv.org/abs/2608.14872
AB - 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.
ER -