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ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T Quantum Mechanics

Authors Emmanuel Jeandel, Simon Perdrix, Renaud Vilmart, Quanlong Wang



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Emmanuel Jeandel
Simon Perdrix
Renaud Vilmart
Quanlong Wang

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Emmanuel Jeandel, Simon Perdrix, Renaud Vilmart, and Quanlong Wang. 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), Volume 83, pp. 11:1-11:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.MFCS.2017.11

Abstract

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.
Keywords
  • Categorical Quantum Mechanincs
  • ZX-Calculus
  • Completeness
  • Cyclotomic Supplmentarity
  • Clifford+T

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References

  1. Miriam Backens. The zx-calculus is complete for stabilizer quantum mechanics. New Journal of Physics, 16(9):093021, 2014. URL: http://dx.doi.org/10.1088/1367-2630/16/9/093021.
  2. Miriam Backens. The zx-calculus is complete for the single-qubit clifford+t group. Electronic Proceedings in Theoretical Computer Science, 2014. URL: http://dx.doi.org/10.4204/EPTCS.172.21.
  3. Miriam Backens. Making the stabilizer zx-calculus complete for scalars. Electronic Proceedings in Theoretical Computer Science, 2015. URL: http://dx.doi.org/10.4204/EPTCS.195.2.
  4. Miriam Backens and Ali Nabi Duman. A complete graphical calculus for spekkens' toy bit theory. Foundations of Physics, pages 1-34, 2014. URL: http://dx.doi.org/10.1007/s10701-015-9957-7.
  5. Miriam Backens, Simon Perdrix, and Quanlong Wang. A simplified stabilizer zx-calculus. Electronic Proceedings in Theoretical Computer Science, 2016. URL: http://dx.doi.org/10.4204/EPTCS.236.1.
  6. Categorical quantum mechanics: Zx-completeness. URL: http://cqm.wikidot.com/zx-completeness.
  7. Bob Coecke. Axiomatic description of mixed states from selinger’s cpm-construction. Electron. Notes Theor. Comput. Sci., 210:3-13, July 2008. URL: http://dx.doi.org/10.1016/j.entcs.2008.04.014.
  8. Bob Coecke and Ross Duncan. Interacting quantum observables: categorical algebra and diagrammatics. New Journal of Physics, 13(4):043016, 2011. URL: http://dx.doi.org/10.1088/1367-2630/13/4/043016.
  9. Bob Coecke and Bill Edwards. Three qubit entanglement within graphical z/x-calculus. Electronic Proceedings in Theoretical Computer Science, 52:22-33, 2011. URL: http://dx.doi.org/10.4204/EPTCS.52.3.
  10. Bob Coecke and Simon Perdrix. Environment and classical channels in categorical quantum mechanics. Logical Methods in Computer Science, Volume 8, Issue 4, November 2012. URL: http://dx.doi.org/10.2168/LMCS-8(4:14)2012.
  11. Ross Duncan and Simon Perdrix. Graphs states and the necessity of euler decomposition. Mathematical Theory and Computational Practice, 5635:167-177, 2009. URL: http://dx.doi.org/10.1007/978-3-642-03073-4.
  12. Ross Duncan and Simon Perdrix. Rewriting measurement-based quantum computations with generalised flow. Lecture Notes in Computer Science, 6199:285-296, 2010. URL: http://dx.doi.org/10.1007/978-3-642-14162-1_24.
  13. Ross Duncan and Simon Perdrix. Pivoting makes the zx-calculus complete for real stabilizers. Electronic Proceedings in Theoretical Computer Science, 2013. URL: http://dx.doi.org/10.4204/EPTCS.171.5.
  14. Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. A complete axiomatisation of the zx-calculus for clifford+ t quantum mechanics. arXiv preprint arXiv:1705.11151, 2017. Google Scholar
  15. Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. Y-calculus: A language for real matrices derived from the zx-calculus. In Conference on Quantum Physics and Logics (QPL'17), 2017. Google Scholar
  16. Simon Perdrix and Quanlong Wang. Supplementarity is necessary for quantum diagram reasoning. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016), volume 58 of Leibniz International Proceedings in Informatics (LIPIcs), pages 76:1-76:14, Krakow, Poland, August 2016. URL: http://dx.doi.org/10.4230/LIPIcs.MFCS.2016.76.
  17. Christian Schröder de Witt and Vladimir Zamdzhiev. The zx-calculus is incomplete for quantum mechanics. Electronic Proceedings in Theoretical Computer Science, 2014. URL: http://dx.doi.org/10.4204/EPTCS.172.20.
  18. Peter Selinger. Finite dimensional hilbert spaces are complete for dagger compact closed categories. Logical Methods in Computer Science, 8(4):1-12, 2012. URL: http://dx.doi.org/10.2168/LMCS-8 (3:06) 2012.
  19. Peter Selinger. Quantum circuits of t-depth one. Phys. Rev. A, 87:042302, Apr 2013. URL: http://dx.doi.org/10.1103/PhysRevA.87.042302.
  20. Robert Spekkens. Evidence for the epistemic view of quantum states: A toy theory. Phys. Rev. A, 75:032110, Mar 2007. URL: http://dx.doi.org/10.1103/PhysRevA.75.032110.
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