{
  "id": "2405.10896",
  "version": "v1",
  "published": "2024-05-17T16:35:07.000Z",
  "updated": "2024-05-17T16:35:07.000Z",
  "title": "ZX-calculus is Complete for Finite-Dimensional Hilbert Spaces",
  "authors": [
    "Boldizsár Poór",
    "Razin A. Shaikh",
    "Quanlong Wang"
  ],
  "comment": "32 pages",
  "categories": [
    "quant-ph"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-17T16:35:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite-dimensional hilbert spaces",
      "zx-calculus",
      "completeness",
      "quantum information theory",
      "finite-dimensional quantum theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}