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Completeness and the ZX-calculus [article]

Miriam Backens
2016 arXiv   pre-print
I develop a graphical 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  ... 
arXiv:1602.08954v1 fatcat:tgiwn32u5bggpao3wfzen2didm

Supplementarity is Necessary for Quantum Diagram Reasoning [article]

Simon Perdrix, Quanlong Wang
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.  ... 
arXiv:1506.03055v1 fatcat:mcf2plowjvg2lcjx52qeuv4r4q

An algebraic axiomatisation of ZX-calculus [article]

Quanlong Wang
2021 arXiv   pre-print
With this algebraic axiomatisation of 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.  ... 
arXiv:1911.06752v2 fatcat:aawmting4falldeyhck2jbbsfa

The ZX-calculus is incomplete for quantum mechanics

Christian Schröder de Witt, Vladimir Zamdzhiev
2014 Electronic Proceedings in Theoretical Computer Science  
We suggest 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.  ... 
doi:10.4204/eptcs.172.20 fatcat:jwuz456l4zhsjiuwphtbd2wwti

A universal completion of the ZX-calculus [article]

Kang Feng Ng, Quanlong Wang
2017 arXiv   pre-print
In this paper, we give a universal 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.  ... 
arXiv:1706.09877v1 fatcat:pabesylzhbbczjq2cjcn3n3lq4

Completeness of the ZX-calculus for Pure Qubit Clifford+T Quantum Mechanics [article]

Kang Feng Ng, Quanlong Wang
2018 arXiv   pre-print
Recently, we gave a 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.  ... 
arXiv:1801.07993v2 fatcat:3aojuxfzdrgsdbrh4ujwlmhpje

Completeness of the Phase-free ZH-calculus [article]

John van de Wetering, Sal Wolffs
2019 arXiv   pre-print
Our completeness result follows by reducing to Vilmart's rule-set for the phase-free ΔZX-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.  ... 
arXiv:1904.07545v1 fatcat:rbax3qrw5vff7fc5shnyu2xqre

A ZX-Calculus with Triangles for Toffoli-Hadamard, Clifford+T, and Beyond

Renaud Vilmart
2019 Electronic Proceedings in Theoretical Computer Science  
We show that a single axiom is then necessary 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.  ... 
doi:10.4204/eptcs.287.18 fatcat:nj5va3jysfcvnfuvicusni2qrq

A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics [article]

Emmanuel Jeandel, Simon Perdrix, Renaud Vilmart
2018 arXiv   pre-print
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.  ...  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.  ... 
arXiv:1705.11151v2 fatcat:qj3fahvg5ncwrndq4awlohw6ie

Diagrammatic Reasoning beyond Clifford+T Quantum Mechanics [article]

Emmanuel Jeandel, Simon Perdrix, Renaud Vilmart
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.  ... 
arXiv:1801.10142v2 fatcat:5plshrr6mnfrhpytugn5zzcfwu

Diagrammatic Reasoning beyond Clifford+T Quantum Mechanics

Emmanuel Jeandel, Simon Perdrix, Renaud Vilmart
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.  ... 
doi:10.1145/3209108.3209139 dblp:conf/lics/JeandelPV18a fatcat:n3phfonktjeyte66xvl5cxhjxm

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

Emmanuel Jeandel, Simon Perdrix, Renaud Vilmart
2018 Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science - LICS '18  
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.  ...  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.  ... 
doi:10.1145/3209108.3209131 dblp:conf/lics/JeandelPV18 fatcat:eoixhygq7nfsraj25qctolozeu

A Simplified Stabilizer ZX-calculus

Miriam Backens, Simon Perdrix, Quanlong Wang
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.  ... 
doi:10.4204/eptcs.236.1 fatcat:esbg4zsi4vcdbgy5pqc3nncioy

Making the stabilizer ZX-calculus complete for scalars

Miriam Backens
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  ... 
doi:10.4204/eptcs.195.2 fatcat:vymqlbkcrjglxlke7im44cloz4

Pivoting makes the ZX-calculus complete for real stabilizers

Ross Duncan, Simon Perdrix
2014 Electronic Proceedings in Theoretical Computer Science  
We derive an angle-free version of 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.  ... 
doi:10.4204/eptcs.171.5 fatcat:jxubdy4nnbbzhh4pimdnav6bda
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