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.

We show that several problems which are known to be undecidable for probabilistic automata become decidable for quantum finite automata. Our main tool is an algebraic result of independent interest: we give an algorithm which, given a finite number of invertible matrices, computes the Zariski closure of the group generated by these matrices. RésuméNous montrons ici que plusieurs problèmes indécidables pour des automates probabilistes sont décidables pour des automates quantiques. Ce rsultat s'appuie sur un algorithme intéressant en soi, qui,étant donné des matrices inversibles, calcule la cloture de Zariski du groupe engendré par ses matrices.Abstract. We show that several problems which are known to be undecidable for probabilistic automata become decidable for quantum finite automata. Our main tool is an algebraic result of independent interest: we give an algorithm which, given a finite number of invertible matrices, computes the Zariski closure of the group generated by these matrices.Harm Derksen is partially supported by NSF, grant DMS 0102193.

We study the following decision problem: is the language recognized by a quantum finite automaton empty or non-empty? We prove that this problem is decidable or undecidable depending on whether recognition is defined by strict or non-strict thresholds. This result is in contrast with the corresponding situation for probabilistic finite automata for which it is known that strict and non-strict thresholds both lead to undecidable problems.

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.

A [Wang tile](https://en.wikipedia.org/wiki/Wang_tile) is a square tile such that each of its edges is colored. The plane can be _tiled_ with a set of Wang tiles if tiles contained in the set can be placed in the plane without rotations and reflections such that the whole plane is covered and the colors of their edges match at adjacent tiles. Wang tiles were introduced by [Wang](https://en.wikipedia.org/wiki/Hao_Wang_(academic)) in 1961 to study decidability in mathematical logic, and they are also of relevance to other areas of theoretical computer science. Wang conjectured that if the plane can be tiled with a set of Wang tiles, then it can be tiled in a periodic way. This was refuted by [Berger](https://en.wikipedia.org/wiki/Robert_Berger_(mathematician)) in 1966 who described how a Turing machine computation can be emulated by Wang tilings and constructed a set of 104 Wang tiles for which the plane can be tiled with the set but only aperiodically. In the first volume of [The Art of Computer Programming](https://en.wikipedia.org/wiki/The_Art_of_Computer_Programming), Knuth presented a simplified version of Berger's set with 92 Wang tiles. Smaller sets of Wang tiles that tile the plane only aperiodically were subsequently constructed with the smallest set containing 13 Wang tiles, found by Culik II in 1996. The authors construct a set of 11 Wang tiles with edges colored with four colors such that the plane can be tiled with the set but each tiling is aperiodic. Moreover, they establish that there is no set of at most 10 Wang tiles with this property. The number of colors is also the best possible as it is known that every set of Wang tiles with edges colored with at most three colors either can tile the plane periodically or cannot tile the plane at all.

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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.