Renaud Vilmard - Completeness of the ZX-Calculus for Quantum Computing

The ZX-Calculus, introduced in 2018 by Bob Coecke and Ross Duncan, is a powerful language for quantum mechanics. Its objects are open graphs that represent quantum evolutions, and it comes with a set of axioms: a set of local graph transformations that are allowed, for they preserve the semantics. The language has numerous applications, amongst which the study of measurement based quantum computing, quantum error correction, verification, quantum protocols, compilation. It is also worth noting that it is closely related to quantum circuits, and that it handily represents all the operations performed in lattice surgery.

A major concern about this language, and an impediment to its broader use, was for some time the question of completeness. Completeness is a particular property that states that any two diagrams that represent the same quantum evolution can be turned into one another by mere application of the axioms. Conceptually, it means that the language captures the whole of quantum computing, and, with it, one can reason about quantum processes solely using the ZX-Calculus, and not the usual formalism of Hilbert spaces.

During this talk, I will present some of the result I got during my Ph.D. thesis on the matter of completeness of the ZX-Calculus. We will see in particular two different approaches for tackling the problem.