We introduce the first complete and approximatively universal diagrammatic language for quantum mechanics. We make the ZX-Calculus, a diagrammatic language introduced by Coecke and Duncan, complete for the so-called Clifford+T quantum mechanics by adding two new axioms to the language. The completeness of the ZX-Calculus for Clifford+T quantum mechanics was one of the main open questions in categorical quantum mechanics. We prove the completeness of the π 4 -fragment of the ZX-Calculus using the recently studied ZW-Calculus, a calculus dealing with integer matrices. We also prove that the π 4 -fragment of the ZX-Calculus represents exactly all the matrices over some finite dimensional extension of the ring of dyadic rationals.Part 4 (Corollary 1) Let D 1 , D 2 be a diagram of the ZX-Calculus. If ZX π/4 ⊢ D 1 XW W X = D 2 XW W X then ZX π/4 ⊢ D 1 = D 2 . Our main theorem is now obvious:Proof (Proof of Theorem 1). Let D 1 , D 2 be two diagrams of the ZX π/4 -Calculus s.t. D 1 = D 2 . By Part 2, D 1 XW = D 2 XW . By Part 1, the ZW 1/2 -calculus is complete and therefore ZW 1/2 ⊢ D 1 XW = D 2 XW . By Part 3, ZX π/4 ⊢ D 1 XW W X = D 2 XW W X . By Part 4 this implies ZX π/4 ⊢ D 1 = D 2 .⊓ ⊔ This approach gives a completion procedure. It gives a set of equalities between ZX π/4 -diagrams whose derivability proves the completeness of the language. Hence, the new rules of the ZX π/4 -Calculus we introduced have obviously been chosen for Parts 4 and 3 to hold. However they have been greatly simplified from what one can obtain using the approach naively.

The ZX-Calculus is a graphical language for diagrammatic reasoning in quantum mechanics and quantum information theory. An axiomatisation has recently been proven to be complete for an approximatively universal fragment of quantum mechanics, the so-called Clif-ford+T fragment. We focus here on the expressive power of this axiomatisation beyond Clifford+T Quantum mechanics. We consider the full pure qubit quantum mechanics, and mainly prove two results: (i) First, the axiomatisation for Clifford+T quantum mechanics is also complete for all equations involving some kind of linear diagrams. The linearity of the diagrams reflects the phase group structure, an essential feature of the ZX-calculus. In particular all the axioms of the ZX-calculus are involving linear diagrams. (ii) We also show that the axiomatisation for Clifford+T is not complete in general but can be completed by adding a single (non linear) axiom, providing a simpler axiomatisation of the ZXcalculus for pure quantum mechanics than the one recently introduced by Ng&Wang.

Recent developments in the ZX-Calculus have resulted in complete axiomatisations first for an approximately universal restriction of the language, and then for the whole language. The main drawbacks were that the axioms that were added to achieve completeness were numerous, tedious to manipulate and lacked a physical interpretation. We present in this paper two complete axiomatisations for the general ZX-Calculus, that we believe are optimal, in that all their equations are necessary and moreover have a nice physical interpretation.

Recent completeness results on the ZX-Calculus used a third-party language, namely the ZW-Calculus. As a consequence, these proofs are elegant, but sadly non-constructive. We address this issue in the following. To do so, we first describe a generic normal form for ZX-diagrams in any fragment that contains Clifford+T quantum mechanics. We give sufficient conditions for an axiomatisation to be complete, and an algorithm to reach the normal form. Finally, we apply these results to the Clifford+T fragment and the general ZX-Calculus -for which we already know the completeness-, but also for any fragment of rational angles: we show that the axiomatisation for Clifford+T is also complete for any fragment of dyadic angles, and that a simple new rule (called cancellation) is necessary and sufficient otherwise.

There exist several graphical languages for quantum information processing, like quantum circuits, ZX-calculus, ZW-calculus, and so on. Each of these languages forms a †-symmetric monoidal category (†-SMC) and comes with an interpretation functor to the †-SMC of finite-dimensional Hilbert spaces. In recent years, one of the main achievements of the categorical approach to quantum mechanics has been to provide several equational theories for most of these graphical languages, making them complete for various fragments of pure quantum mechanics. We address the question of how to extend these languages beyond pure quantum mechanics to reason about mixed states and general quantum operations, i.e., completely positive maps. Intuitively, such an extension relies on the axiomatisation of a discard map that allows one to get rid of a quantum system, an operation that is not allowed in pure quantum mechanics. We introduce a new construction, the discard construction , which transforms any †-symmetric monoidal category into a symmetric monoidal category equipped with a discard map. Roughly speaking this construction consists in making any isometry causal. Using this construction, we provide an extension for several graphical languages that we prove to be complete for general quantum operations. However, this construction fails for some fringe cases like Clifford+T quantum mechanics, as the category does not have enough isometries.

We show that the formalism of “Sum-Over-Path” (SOP), used for symbolically representing linear maps or quantum operators, together with a proper rewrite system, has the structure of a dagger-compact PROP. Several consequences arise from this observation:– Morphisms of SOP are very close to the diagrams of the graphical calculus called ZH-Calculus, so we give a system of interpretation between the two– A construction, called the discard construction, can be applied to enrich the formalism so that, in particular, it can represent the quantum measurement.We also enrich the rewrite system so as to get the completeness of the Clifford fragments of both the initial formalism and its enriched version.

We consider a ZX-calculus augmented with triangle nodes which is well-suited to reason on the so-called Toffoli-Hadamard fragment of quantum mechanics. We precisely show the form of the matrices it represents, and we provide an axiomatisation which makes the language complete for the Toffoli-Hadamard quantum mechanics. We extend the language with arbitrary angles and show that any true equation involving linear diagrams which constant angles are multiple of π are derivable. We show that a single axiom is then necessary and sufficient to make the language equivalent to the ZX-calculus which is known to be complete for Clifford+T quantum mechanics. As a by-product, it leads to a new and simple complete axiomatisation for Clifford+T quantum mechanics.

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