ZX-calculus is Complete for Finite-Dimensional Hilbert Spaces
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866915452263858176 |
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| author | Poór, Boldizsár Shaikh, Razin A. Wang, Quanlong |
| author_facet | Poór, Boldizsár Shaikh, Razin A. Wang, Quanlong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_10896 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | ZX-calculus is Complete for Finite-Dimensional Hilbert Spaces Poór, Boldizsár Shaikh, Razin A. Wang, Quanlong Quantum Physics 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. |
| title | ZX-calculus is Complete for Finite-Dimensional Hilbert Spaces |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2405.10896 |