45 research outputs found
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
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 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
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
Pivoting makes the ZX-calculus complete for real stabilizers
We show that pivoting property of graph states cannot be derived from the
axioms of the ZX-calculus, and that pivoting does not imply local
complementation of graph states. Therefore the ZX-calculus augmented with
pivoting is strictly weaker than the calculus augmented with the Euler
decomposition of the Hadamard gate. We derive an angle-free version of the
ZX-calculus and show that it is complete for real stabilizer quantum mechanics.Comment: In Proceedings QPL 2013, arXiv:1412.791
Qutrit ZX-calculus is Complete for Stabilizer Quantum Mechanics
In this paper, we show that a qutrit version of ZX-calculus, with rules
significantly different from that of the qubit version, is complete for pure
qutrit stabilizer quantum mechanics, where state preparations and measurements
are based on the three dimensional computational basis, and unitary operations
are required to be in the generalized Clifford group. This means that any
equation of diagrams that holds true under the standard interpretation in
Hilbert spaces can be derived diagrammatically. In contrast to the qubit case,
the situation here is more complicated due to the richer structure of this
qutrit ZX-calculus.Comment: In Proceedings QPL 2017, arXiv:1802.0973
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