381 research outputs found
The rational fragment of the ZX-calculus
We introduce here a new axiomatisation of the rational fragment of the
ZX-calculus, a diagrammatic language for quantum mechanics. Compared to the
previous axiomatisation introduced in [8], our axiomatisation does not use any
metarule , but relies instead on a more natural rule, called the cyclotomic
supplementarity rule, that was introduced previously in the literature. Our
axiomatisation is only complete for diagrams using rational angles , and is not
complete in the general case. Using results on diophantine geometry, we
characterize precisely which diagram equality involving arbitrary angles are
provable in our framework without any new axioms, and we show that our
axiomatisation is continuous, in the sense that a diagram equality involving
arbitrary angles is provable iff it is a limit of diagram equalities involving
rational angles. We use this result to give a complete characterization of all
Euler equations that are provable in this axiomatisation
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
The rational fragment of the ZX-calculus
We introduce here a new axiomatisation of the rational fragment of the ZX-calculus, a diagrammatic language for quantum mechanics. Compared to the previous axiomatisation introduced in [8], our axiomatisation does not use any metarule , but relies instead on a more natural rule, called the cyclotomic supplementarity rule, that was introduced previously in the literature. Our axiomatisation is only complete for diagrams using rational angles , and is not complete in the general case. Using results on diophantine geometry, we characterize precisely which diagram equality involving arbitrary angles are provable in our framework without any new axioms, and we show that our axiomatisation is continuous, in the sense that a diagram equality involving arbitrary angles is provable iff it is a limit of diagram equalities involving rational angles. We use this result to give a complete characterization of all Euler equations that are provable in this axiomatisation
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
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
ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T quantum mechanics
The ZX-Calculus is a powerful graphical language for quantum mechanics and
quantum information processing. The completeness of the language -- i.e. the
ability to derive any true equation -- is a crucial question. In the quest of a
complete ZX-calculus, supplementarity has been recently proved to be necessary
for quantum diagram reasoning (MFCS 2016). Roughly speaking, supplementarity
consists in merging two subdiagrams when they are parameterized by antipodal
angles. We introduce a generalised supplementarity -- called cyclotomic
supplementarity -- which consists in merging n subdiagrams at once, when the n
angles divide the circle into equal parts. We show that when n is an odd prime
number, the cyclotomic supplementarity cannot be derived, leading to a
countable family of new axioms for diagrammatic quantum reasoning.We exhibit
another new simple axiom that cannot be derived from the existing rules of the
ZX-Calculus, implying in particular the incompleteness of the language for the
so-called Clifford+T quantum mechanics. We end up with a new axiomatisation of
an extended ZX-Calculus, including an axiom schema for the cyclotomic
supplementarity.Comment: Mathematical Foundations of Computer Science, Aug 2017, Aalborg,
Denmar
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
Supplementarity is Necessary for Quantum Diagram Reasoning
The ZX-calculus is a powerful diagrammatic language for quantum mechanics and
quantum information processing. We prove that its \pi/4-fragment is not
complete, in other words the ZX-calculus is not complete for the so called
"Clifford+T quantum mechanics". The completeness of this fragment was one of
the main open problems in categorical quantum mechanics, a programme initiated
by Abramsky and Coecke. The ZX-calculus was known to be incomplete for quantum
mechanics. On the other hand, its \pi/2-fragment is known to be complete, i.e.
the ZX-calculus is complete for the so called "stabilizer quantum mechanics".
Deciding whether its \pi/4-fragment is complete is a crucial step in the
development of the ZX-calculus since this fragment is approximately universal
for quantum mechanics, contrary to the \pi/2-fragment. To establish our
incompleteness result, we consider a fairly simple property of quantum states
called supplementarity. We show that supplementarity can be derived in the
ZX-calculus if and only if the angles involved in this equation are multiples
of \pi/2. In particular, the impossibility to derive supplementarity for \pi/4
implies the incompleteness of the ZX-calculus for Clifford+T quantum mechanics.
As a consequence, we propose to add the supplementarity to the set of rules of
the ZX-calculus. We also show that if a ZX-diagram involves antiphase twins,
they can be merged when the ZX-calculus is augmented with the supplementarity
rule. Merging antiphase twins makes diagrammatic reasoning much easier and
provides a purely graphical meaning to the supplementarity rule.Comment: Generalised proof and graphical interpretation. 16 pages, submitte
Towards a Minimal Stabilizer ZX-calculus
The stabilizer ZX-calculus is a rigorous graphical language for reasoning
about quantum mechanics. The language is sound and complete: one can transform
a stabilizer ZX-diagram into another one using the graphical rewrite rules if
and only if these two diagrams represent the same quantum evolution or quantum
state. We previously showed that the stabilizer ZX-calculus can be simplified
by reducing the number of rewrite rules, without losing the property of
completeness [Backens, Perdrix & Wang, EPTCS 236:1--20, 2017]. Here, we show
that most of the remaining rules of the language are indeed necessary. We do
however leave as an open question the necessity of two rules. These include,
surprisingly, the bialgebra rule, which is an axiomatisation of
complementarity, the cornerstone of the ZX-calculus. Furthermore, we show that
a weaker ambient category -- a braided autonomous category instead of the usual
compact closed category -- is sufficient to recover the meta rule 'only
connectivity matters', even without assuming any symmetries of the generators.Comment: 29 pages, minor updates for v
A Diagrammatic Axiomatisation for Qubit Entanglement
Diagrammatic techniques for reasoning about monoidal categories provide an
intuitive understanding of the symmetries and connections of interacting
computational processes. In the context of categorical quantum mechanics,
Coecke and Kissinger suggested that two 3-qubit states, GHZ and W, may be used
as the building blocks of a new graphical calculus, aimed at a diagrammatic
classification of multipartite qubit entanglement that would highlight the
communicational properties of quantum states, and their potential uses in
cryptographic schemes.
In this paper, we present a full graphical axiomatisation of the relations
between GHZ and W: the ZW calculus. This refines a version of the preexisting
ZX calculus, while keeping its most desirable characteristics: undirectedness,
a large degree of symmetry, and an algebraic underpinning. We prove that the ZW
calculus is complete for the category of free abelian groups on a power of two
generators - "qubits with integer coefficients" - and provide an explicit
normalisation procedure.Comment: 12 page
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