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A complete axiomatisation of the ZX-calculus for Clifford+T quantum mechanics. (English) Zbl 1452.81080

Proceedings of the 2018 33rd annual ACM/IEEE symposium on logic in computer science, LICS 2018, Oxford, UK, July 9–12, 2018. New York, NY: Association for Computing Machinery (ACM). 559-568 (2018).

MSC:

81P68 Quantum computation
18M30 String diagrams and graphical calculi
81P65 Quantum gates
15A67 Applications of Clifford algebras to physics, etc.

References:

[1] Scott Aaronson and Daniel Gottesman. 2004. Improved simulation of stabilizer circuits. Phys. Rev. A 70 (Nov 2004), 052328. Issue 5.
[2] Miriam Backens. 2014. The ZX-calculus is complete for stabilizer quantum mechanics. New Journal of Physics 16, 9 (2014), 093021. http://stacks.iop.org/1367-2630/16/i=9/a=093021 · Zbl 1451.81014
[3] Miriam Backens. 2014. The ZX-calculus is complete for the single-qubit Clifford+T group. Electronic Proceedings in Theoretical Computer Science 172 (2014), 293-303. arXiv:1412.8553 · Zbl 1467.81025
[4] Miriam Backens and Ali Nabi Duman. 2016. A Complete Graphical Calculus for Spekkens’ Toy Bit Theory. Foundations of Physics 46, 1 (01 Jan 2016), 70-103. · Zbl 1347.81019
[5] Miriam Backens, Simon Perdrix, and Quanlong Wang. 2017. Towards a Minimal Stabilizer ZX-calculus. (Sept. 2017). https://arxiv.org/abs/1709.08903 · Zbl 1484.18017
[6] Nicholas Chancellor, Aleks Kissinger, Joschka Roffe, Stefan Zohren, and Dominic Horsman. 2016. Graphical Structures for Design and Verification of Quantum Error Correction. (2016). https://arxiv.org/abs/1611.08012 · Zbl 1433.94132
[7] Bob Coecke and Ross Duncan. 2011. Interacting quantum observables: categorical algebra and diagrammatics. New Journal of Physics 13, 4 (2011), 043016. http://stacks.iop.org/1367-2630/13/i=4/a=043016 · Zbl 1448.81025
[8] Bob Coecke and Aleks Kissinger. 2010. The Compositional Structure of Multipartite Quantum Entanglement. Springer Berlin Heidelberg, Berlin, Heidelberg, 297-308. · Zbl 1288.81025
[9] Bob Coecke and Aleks Kissinger. 2017. Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning. Cambridge University Press, Cambridge. · Zbl 1405.81001
[10] Niel de Beaudrap and Dominic Horsman. 2017. The ZX calculus is a language for surface code lattice surgery. (2017). arXiv:1704.08670 https://arxiv.org/abs/1704.08670
[11] Ross Duncan. 2013. A graphical approach to measurement-based quantum computing. Oxford University Press, Oxford, Chapter 3, 50-89. · Zbl 1317.81059
[12] Ross Duncan and Kevin Dunne. 2016. Interacting Frobenius Algebras Are Hopf. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2016). ACM, New York, NY, USA, 535-544. 10.1145/2933575.2934550 · Zbl 1392.68297
[13] Ross Duncan and Liam Garvie. 2017. Verifying the Smallest Interesting Colour Code with Quantomatic. (2017). https://arxiv.org/abs/1706.02717 · Zbl 1486.81072
[14] Ross Duncan and Maxime Lucas. 2014. Verifying the Steane code with Quantomatic. Electronic Proceedings in Theoretical Computer Science 171 (2014), 33-49. · Zbl 1464.81025
[15] Ross Duncan and Simon Perdrix. 2010. Rewriting measurement-based quantum computations with generalised flow. Lecture Notes in Computer Science 6199 (2010), 285-296. · Zbl 1288.68069
[16] Ross Duncan and Simon Perdrix. 2013. Pivoting makes the ZX-calculus complete for real stabilizers, In QPL 2013. Electronic Proceedings in Theoretical Computer Science 171, 50-62. arXiv:1307.7048 · Zbl 1467.81050
[17] Brett Giles and Peter Selinger. 2013. Exact synthesis of multiqubit Clifford+T circuits. Phys. Rev. A 87 (Mar 2013), 032332. Issue 3.
[18] Amar Hadzihasanovic. 2015. A Diagrammatic Axiomatisation for Qubit Entanglement. In 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science. ACM/IEEE, Kyoto, 573-584. 10.1109/LICS.2015.59 · Zbl 1401.81012
[19] Amar Hadzihasanovic. 2017. The algebra of entanglement and the geometry of composition. Ph.D. Dissertation. University of Oxford. https://arxiv.org/abs/1709.08086 · Zbl 1401.81012
[20] Clare Horsman. 2011. Quantum picturalism for topological cluster-state computing. New Journal of Physics 13, 9 (2011), 095011. http://stacks.iop.org/1367-2630/13/i=9/a=095011 · Zbl 1448.81239
[21] Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. 2017. A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics. (2017). https://arxiv.org/abs/1705.11151 · Zbl 1452.81080
[22] Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. 2018. Diagrammatic Reasoning beyond Clifford+T Quantum Mechanics. (2018). https://arxiv.org/abs/1801.10142 · Zbl 1452.81081
[23] Emmanuel Jeandel, Simon Perdrix, Renaud Vilmart, and Quanlong Wang. 2017. ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T Quantum Mechanics. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017) (Leibniz International Proceedings in Informatics (LIPIcs)), Kim G. Larsen, Hans L. Bodlaender, and Jean-Francois Raskin (Eds.), Vol. 83. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 11:1-11:13. · Zbl 1447.81077
[24] A. Kissinger, L. Dixon, R. Duncan, B. Frot, A. Merry, D. Quick, M. Soloviev, and V Zamdzhiev. 2011. Quantomatic. (2011). https://sites.google.com/site/quantomatic/
[25] Aleks Kissinger and Vladimir Zamdzhiev. 2015. Quantomatic: A Proof Assistant for Diagrammatic Reasoning. In Automated Deduction - CADE-25, Amy P. Felty and Aart Middeldorp (Eds.). Springer International Publishing, Cham, 326-336. · Zbl 1465.68288
[26] Kang Feng Ng and Quanlong Wang. 2017. A universal completion of the ZX-calculus. (2017). https://arxiv.org/abs/1706.09877
[27] Kang Feng Ng and Quanlong Wang. 2018. Completeness of the ZX-calculus for Pure Qubit Clifford+T Quantum Mechanics. (2018). https://arxiv.org/abs/1801.07993 · Zbl 1452.81079
[28] Simon Perdrix and Quanlong Wang. 2016. Supplementarity is Necessary for Quantum Diagram Reasoning. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016) (Leibniz International Proceedings in Informatics (LIPIcs)), Vol. 58. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Krakow, Poland, 76:1-76:14. · Zbl 1398.81015
[29] Steven Roman. 2006. Field Theory. Springer, New York. · Zbl 1172.12001
[30] Christian Schröder de Witt and Vladimir Zamdzhiev. 2014. The ZX-calculus is incomplete for quantum mechanics, In QPL 2014. Electronic Proceedings in Theoretical Computer Science 172,285-292. arXiv:1404.3633 · Zbl 1467.81028
[31] Yaoyun Shi. 2003. Both Toffoli and controlled-NOT Need Little Help to Do Universal Quantum Computing. Quantum Info. Comput. 3, 1 (Jan. 2003), 84-92. http://dl.acm.org/citation.cfm?id=2011508.2011515 · Zbl 1152.81811
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