The ZX Calculus is incomplete for Clifford+T quantum mechanics

The ZX calculus is a diagrammatic language for quantum mechanics and quantum information processing. We prove that the ZX-calculus is not complete for the Clifford+T quantum mechanics. The completeness for this fragment has been stated as one of the main current open problems in categorical quantum mechanics [8]. The ZX calculus was known to be incomplete for quantum mechanics [7], on the other hand, it has been proved complete for Clifford quantum mechanics (a.k.a. stabilizer quantum mechanics) [1], and for single-qubit Clifford+T quantum mechanics [2]. The question of the completeness of the ZX calculus for Clifford+T is a crucial step in the development of the ZX calculus because of its (approximate) universality for quantum mechanics (i.e. any unitary evolution can be approximated using Clifford and T gates only). We exhibit a property which is know to be true in Clifford+T quantum mechanics and prove that this equation cannot be derived in the ZX calculus, by introducing a new sound interpretation of the ZX calculus in which this particular property does not hold. Finally, we propose to extend the language with a new axiom.