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## Filters

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Completeness and the ZX-calculus
[article]

2016
*
arXiv
*
pre-print

I develop a graphical

arXiv:1602.08954v1
fatcat:tgiwn32u5bggpao3wfzen2didm
*calculus*similar to*the**ZX*-*calculus*that fully describes Spekkens' toy theory,*and*show that it is*complete*. ... Furthermore, I show that*the**ZX*-*calculus*is*complete*for*the*single-qubit Clifford+T group, which is approximately universal: any single-qubit unitary can be approximated to arbitrary accuracy using only ...*The*Clifford+T group*The*Chapter 4*The**zx*-*calculus**and**completeness*In*the*previous chapter, we introduced*the**zx*-*calculus*notation*and*rewrite rules,*and*argued that this graphical*calculus*for ...##
###
Supplementarity is Necessary for Quantum Diagram Reasoning
[article]

2015
*
arXiv
*
pre-print

*The*

*ZX*-

*calculus*is a powerful diagrammatic language for quantum mechanics

*and*quantum information processing. ... 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". ... Acknowledgement We want to thank Miriam Backens, Bob Coecke, Ross Duncan, Aleks Kissinger, Emmanuel Jeandel,

*and*Vladimir Zamdzhiev for fruitful

*and*stimulating discussions. ...

##
###
An algebraic axiomatisation of ZX-calculus
[article]

2021
*
arXiv
*
pre-print

With this algebraic axiomatisation of

arXiv:1911.06752v2
fatcat:aawmting4falldeyhck2jbbsfa
*ZX*-*calculus*, we are able to establish for*the*first time a simple translation of diagrams from another graphical language called ZH-*calculus**and*to derive all*the*... As a consequence, we have a great benefit that all techniques obtained in ZH-*calculus*can be transplanted to*ZX*-*calculus*, which can not be obtained by just using*the**completeness*of*ZX*-*calculus*. ...*The*author thanks Bob Coecke, Niel de Beaudrap*and*Konstantinos Meichanetzidis for useful discussions on*the*title of this paper. ...##
###
The ZX-calculus is incomplete for quantum mechanics

2014
*
Electronic Proceedings in Theoretical Computer Science
*

We suggest

doi:10.4204/eptcs.172.20
fatcat:jwuz456l4zhsjiuwphtbd2wwti
*the*addition of a new 'color-swap' rule, of which currently no analytical formulation is known*and*which we suspect may be necessary, but not sufficient to make*the**ZX*-*calculus**complete*. ... We prove that*the**ZX*-*calculus*is incomplete for quantum mechanics. ... One of*the*authors also wishes to gratefully acknowledge support from*the*EPSRC*and**the*Scatcherd European Scholarship. ...##
###
A universal completion of the ZX-calculus
[article]

2017
*
arXiv
*
pre-print

In this paper, we give a universal

arXiv:1706.09877v1
fatcat:pabesylzhbbczjq2cjcn3n3lq4
*completion*of*the**ZX*-*calculus*for*the*whole of pure qubit quantum mechanics. ... This proof is based on*the**completeness*of another graphical language:*the*ZW-*calculus*, with direct translations between these two graphical systems. ... Acknowledgement*The*authors would like to thank Bob Coecke*and*Amar Hadzihasanovic for*the*fruitful discussions*and*invaluable comments. ...##
###
Completeness of the ZX-calculus for Pure Qubit Clifford+T Quantum Mechanics
[article]

2018
*
arXiv
*
pre-print

Recently, we gave a

arXiv:1801.07993v2
fatcat:3aojuxfzdrgsdbrh4ujwlmhpje
*complete*axiomatisation of*the**ZX*-*calculus*for*the*overall pure qubit quantum mechanics. ... Based on this result, here we also obtain a*complete*axiomatisation of*the**ZX*-*calculus*for*the*Clifford+T quantum mechanics by restricting*the*ring of complex numbers to its subring corresponding to*the*... Acknowledgement*The*authors would like to thank Bob Coecke for*the*fruitful discussions*and*invaluable comments. ...##
###
Completeness of the Phase-free ZH-calculus
[article]

2019
*
arXiv
*
pre-print

Our

arXiv:1904.07545v1
fatcat:rbax3qrw5vff7fc5shnyu2xqre
*completeness*result follows by reducing to Vilmart's rule-set for*the*phase-free Δ*ZX*-*calculus*. ... We present a modified rule-set for*the*phase-free ZH-*calculus**and*show that it is*complete*. We further discuss*the*minimality of this rule-set*and*we give an intuitive interpretation of*the*rules. ... This rule bridges*the*gap between*the*ZH-*calculus**and**the**ZX*-*calculus*augmented with this triangle generator. As a result, we can use*the**completeness*of Ref. ...##
###
A ZX-Calculus with Triangles for Toffoli-Hadamard, Clifford+T, and Beyond

2019
*
Electronic Proceedings in Theoretical Computer Science
*

We show that a single axiom is then necessary

doi:10.4204/eptcs.287.18
fatcat:nj5va3jysfcvnfuvicusni2qrq
*and*sufficient to make*the*language equivalent to*the**ZX*-*calculus*which is known to be*complete*for Clifford+T quantum mechanics. ... 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. ... All diagrams were written with*the*help of TikZit.*The*author would like to thank Emmanuel Jeandel*and*Simon Perdrix for fruitful discussions*and*suggestions. ...##
###
A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics
[article]

2018
*
arXiv
*
pre-print

We make

arXiv:1705.11151v2
fatcat:qj3fahvg5ncwrndq4awlohw6ie
*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. ... 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. ... Acknowledgements*The*authors acknowledge support from*the*projects ANR-17-CE25-0009 SoftQPro, ANR-17-CE24-0035 VanQuTe, PIA-GDN/Quantex,*and*STIC-AmSud 16-STIC-05 FoQCoSS. ...##
###
Diagrammatic Reasoning beyond Clifford+T Quantum Mechanics
[article]

2018
*
arXiv
*
pre-print

*The*

*ZX*-

*Calculus*is a graphical language for quantum mechanics. ...

*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. ... Acknowledgements

*The*authors acknowledge support from

*the*projects ANR-17-CE25-0009 Soft-QPro, ANR-17-CE24-0035 VanQuTe, PIA-GDN/Quantex,

*and*STIC-AmSud 16-STIC-05 FoQCoSS. ...

##
###
Diagrammatic Reasoning beyond Clifford+T Quantum Mechanics

2018
*
Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science - LICS '18
*

*The*

*ZX*-

*Calculus*is a graphical language for diagrammatic reasoning in quantum mechanics

*and*quantum information theory. ...

*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. ... Acknowledgements

*The*authors acknowledge support from

*the*projects ANR-17-CE25-0009 Soft-QPro, ANR-17-CE24-0035 VanQuTe, PIA-GDN/Quantex,

*and*STIC-AmSud 16-STIC-05 FoQCoSS. ...

##
###
A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics

2018
*
Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science - LICS '18
*

Acknowledgements

doi:10.1145/3209108.3209131
dblp:conf/lics/JeandelPV18
fatcat:eoixhygq7nfsraj25qctolozeu
*The*authors acknowledge support from*the*projects ANR-17-CE25-0009 SoftQPro, ANR-17-CE24-0035 VanQuTe, PIA-GDN/Quantex,*and*STIC-AmSud 16-STIC-05 FoQCoSS. ... All diagrams were written with*the*help of TikZit. ... Their approach consists in deriving*the**completeness*of*the**ZX*-*calculus*from*the**completeness*of*the*ZW C -*calculus*, using a*completion*procedure based on*the*back*and*forth interpretations. ...##
###
A Simplified Stabilizer ZX-calculus

2017
*
Electronic Proceedings in Theoretical Computer Science
*

*The*stabilizer

*ZX*-

*calculus*is a rigorous graphical language for reasoning about quantum mechanics.The language is sound

*and*

*complete*: a stabilizer

*ZX*-diagram can be transformed into another one if

*and*... We show that

*the*stabilizer

*ZX*-

*calculus*can be simplified, removing unnecessary equations while keeping only

*the*essential axioms which potentially capture fundamental structures of quantum mechanics. ... Acknowledgements

*The*authors would like to thank Bob Coecke, Ross Duncan, Emmanuel Jeandel,

*and*Aleks Kissinger for valuable discussions. ...

##
###
Making the stabilizer ZX-calculus complete for scalars

2015
*
Electronic Proceedings in Theoretical Computer Science
*

*The*

*ZX*-

*calculus*is a graphical language for quantum processes with built-in rewrite rules. ... Here, we replace those scalar-free rewrite rules with correctly scaled ones

*and*show that, by adding one new diagram element

*and*a new rewrite rule,

*the*

*calculus*can be made

*complete*for pure qubit stabilizer ... Acknowledgments I would like to thank Aleks Kissinger for pointing out

*the*issue of zero diagram

*completeness*

*and*for constructive discussion on how to resolve it,

*and*Dominic Horsman for helpful comments ...

##
###
Pivoting makes the ZX-calculus complete for real stabilizers

2014
*
Electronic Proceedings in Theoretical Computer Science
*

We derive an angle-free version of

doi:10.4204/eptcs.171.5
fatcat:jxubdy4nnbbzhh4pimdnav6bda
*the**ZX*-*calculus**and*show that it is*complete*for real stabilizer quantum mechanics. ... 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. ... This work has been partially funded by*the*ANR-10-JCJC-0208 CausaQ grant. ...
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