ZX-calculus is Complete for Finite-Dimensional Hilbert Spaces

• URL: http://arxiv.org/abs/2405.10896v1
• Date: Fri, 17 May 2024 16:35:07 GMT
• Title: ZX-calculus is Complete for Finite-Dimensional Hilbert Spaces
• Authors: Boldizsár Poór, Razin A. Shaikh, Quanlong Wang,
• Abstract summary: The ZX-calculus is a graphical language for quantum computing and quantum information theory. We prove 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.
• Score: 0.09831489366502298
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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