{
  "id": "1903.06035",
  "version": "v1",
  "published": "2019-03-13T16:45:40.000Z",
  "updated": "2019-03-13T16:45:40.000Z",
  "title": "Completeness of the ZX-Calculus",
  "authors": [
    "Emmanuel Jeandel",
    "Simon Perdrix",
    "Renaud Vilmart"
  ],
  "comment": "Contains an appendix. arXiv admin note: text overlap with arXiv:1801.10142",
  "categories": [
    "quant-ph",
    "cs.LO"
  ],
  "abstract": "The ZX-Calculus is a graphical language for diagrammatic reasoning in quantum mechanics and quantum information theory. It comes equipped with an equational presentation. We focus here on a very important property of the language: completeness, which roughly ensures the equational theory captures all of quantum mechanics. We first improve on the known-to-be-complete presentation or the so-called Clifford fragment of the language - a restriction that is not universal - by adding some axioms. Thanks to a system of back-and-forth translation between the ZX-Calculus and a third-party complete graphical language, we prove that the provided axiomatisation is complete for the first approximately universal fragment of the language, namely Clifford+T. We then prove that the expressive power of this presentation, though aimed at achieving completeness for the aforementioned restriction, extends beyond Clifford+T, to a class of diagrams that we call linear with Clifford+T constants. We use another version of the third-party language - and an adapted system of back-and-forth translation - to complete the language for the ZX-Calculus as a whole, that is, with no restriction. We briefly discuss the added axioms, and finally, we provide a complete axiomatisation for an altered version of the language which involves an additional generator, making the presentation simpler.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-13T16:45:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "zx-calculus",
      "completeness",
      "quantum mechanics",
      "back-and-forth translation",
      "first approximately universal fragment"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}