ZX-calculus is Complete for Finite-Dimensional Hilbert Spaces
Abstract
The ZX-calculus is a graphical language for reasoning about quantum computing and quantum information theory. As a complete graphical language, it incorporates a set of axioms rich enough to derive any equation of the underlying formalism. While completeness of the ZX-calculus has been established for qubits and the Clifford fragment of prime-dimensional qudits, universal completeness beyond two-level systems has remained unproven until now. In this paper, we present a proof establishing the completeness of finite-dimensional ZX-calculus, incorporating only the mixed-dimensional Z-spider and the qudit X-spider as generators. Our approach builds on the completeness of another graphical language, the finite-dimensional ZW-calculus, with direct translations between these two calculi. By proving its completeness, we lay a solid foundation for the ZX-calculus as a versatile tool not only for quantum computation but also for various fields within finite-dimensional quantum theory.
Cite
@article{arxiv.2405.10896,
title = {ZX-calculus is Complete for Finite-Dimensional Hilbert Spaces},
author = {Boldizsár Poór and Razin A. Shaikh and Quanlong Wang},
journal= {arXiv preprint arXiv:2405.10896},
year = {2025}
}
Comments
In Proceedings QPL 2025, arXiv:2508.13619