29 research outputs found
A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics
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 four 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
Clifford+T fragment of the ZX-Calculus using the recently studied ZW-Calculus,
a calculus dealing with integer matrices. We also prove that the Clifford+T
fragment of the ZX-Calculus represents exactly all the matrices over some
finite dimensional extension of the ring of dyadic rationals
Completeness of the ZX-Calculus
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
for 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
Diagrammatic Reasoning beyond Clifford+T Quantum Mechanics
The ZX-Calculus is a graphical language for quantum mechanics. An
axiomatisation has recently been proven to be complete for an approximatively
universal fragment of quantum mechanics, the so-called Clifford+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 ZX-calculus for pure quantum mechanics than the one recently introduced by
Ng&Wang
A Generic Normal Form for ZX-Diagrams and Application to the Rational Angle Completeness
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
Completeness of Graphical Languages for Mixed States Quantum Mechanics
There exist several graphical languages for quantum information processing, like quantum circuits, ZX-Calculus, ZW-Calculus, etc. Each of these languages forms a dagger-symmetric monoidal category (dagger-SMC) and comes with an interpretation functor to the dagger-SMC of (finite dimension) Hilbert spaces. In the 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 the extension of these languages beyond pure quantum mechanics, in order to reason on mixed states and general quantum operations, i.e. completely positive maps. Intuitively, such an extension relies on the axiomatisation of a discard map which allows one to get rid of a quantum system, operation which is not allowed in pure quantum mechanics.
We introduce a new construction, the discard construction, which transforms any dagger-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 the Clifford+T quantum mechanics, as the category does not have enough isometries
Y-Calculus: A language for real Matrices derived from the ZX-Calculus
International audienceThe ZX-Calculus is a powerful diagrammatic language devoted to represent complex quantum evolutions. But the advantages of quantum computing still exist when working with rebits, and evolutions with real coefficients. Some models explicitly use rebits, but the ZX-Calculus can not handle these evolutions as it is. Hence, we define an alternative language solely dealing with real matrices, with a new set of rules. We show that three of its non-trivial rules are not derivable from the others and we prove that the language is complete for the π/2-fragment. We define a generalisation of the Hadamard node, and exhibit two interpretations from and to the ZX-Calculus, showing the consistency between the two languages
Completeness of the ZX-Calculus
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
for 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
A Generic Normal Form for ZX-Diagrams and Application to the Rational Angle Completeness
International audienceRecent 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
Diagrammatic Reasoning beyond Clifford+T Quantum Mechanics
International audienceThe 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 Clifford+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 ZX-calculus for pure quantum mechanics than the one recently introduced by Ng&Wang