zbMATH — the first resource for mathematics

ZX-calculus: cyclotomic supplementarity and incompleteness for Clifford+T quantum mechanics. (English) Zbl 1447.81077
Larsen, Kim G. (ed.) et al., 42nd international symposium on mathematical foundations of computer science, MFCS 2017, August 21–25, 2017, Aalborg, Denmark. Wadern: Schloss Dagstuhl – Leibniz Zentrum für Informatik. LIPIcs – Leibniz Int. Proc. Inform. 83, Article 11, 13 p. (2017).
Summary: 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 [the second and last authors, LIPIcs – Leibniz Int. Proc. Inform. 58, Article 76, 14 p. (2016; Zbl 1398.81015)]. 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.
For the entire collection see [Zbl 1376.68011].

81P68 Quantum computation
18M30 String diagrams and graphical calculi
81P65 Quantum gates
15A67 Applications of Clifford algebras to physics, etc.
Full Text: DOI
[2] Miriam Backens.The zx-calculus is complete for stabilizer quantum mechanics.\it New \it Journal of Physics, 16(9):093021, 2014. doi:10.1088/1367-2630/16/9/093021. · Zbl 1347.81019
[3] Miriam Backens. The zx-calculus is complete for the single-qubit clifford+t group. \it Elec- \it tronic Proceedings in Theoretical Computer Science, 2014. doi:10.4204/EPTCS.172.21.
[4] Miriam Backens. Making the stabilizer zx-calculus complete for scalars. \it Electronic Pro- \it ceedings in Theoretical Computer Science, 2015. doi:10.4204/EPTCS.195.2.
[5] Miriam Backens and Ali Nabi Duman. A complete graphical calculus for spekkens’ toy bit theory. \it Foundations of Physics, pages 1-34, 2014. doi:10.1007/s10701-015-9957-7. · Zbl 1347.81019
[6] Miriam Backens, Simon Perdrix, and Quanlong Wang. A simplified stabilizer zx-calculus. \it Electronic Proceedings in Theoretical Computer Science, 2016. doi:10.4204/EPTCS.236.1.
[7] Categorical quantum mechanics:Zx-completeness.URL: http://cqm.wikidot.com/ zx-completeness.
[8] Bob Coecke. Axiomatic description of mixed states from selinger’s cpm-construction. \it Elec- \it tron. Notes Theor. Comput. Sci., 210:3-13, July 2008. doi:10.1016/j.entcs.2008.04. 014. · Zbl 1279.81010
[9] Bob Coecke and Ross Duncan. Interacting quantum observables: categorical algebra and diagrammatics. \it New Journal of Physics, 13(4):043016, 2011. doi:10.1088/1367-2630/ 13/4/043016. · Zbl 1155.81316
[10] Bob Coecke and Bill Edwards. Three qubit entanglement within graphical z/x-calculus. \it Electronic Proceedings in Theoretical Computer Science, 52:22-33, 2011. doi:10.4204/ EPTCS.52.3.
[11] Bob Coecke and Simon Perdrix. Environment and classical channels in categorical quantum mechanics. \it Logical Methods in Computer Science, Volume 8, Issue 4, November 2012. doi:10.2168/LMCS-8(4:14)2012. · Zbl 1259.81019
[12] Ross Duncan and Simon Perdrix. Graphs states and the necessity of euler decomposition.\it Mathematical Theory and Computational Practice, 5635:167-177, 2009.doi:10.1007/978-3-642-03073-4. · Zbl 1268.81046
[13] Ross Duncan and Simon Perdrix. Graphs states and the necessity of euler decomposition. \it Mathematical Theory and Computational Practice, 5635:167-177, 2009.doi:10.1007/ 978-3-642-03073-4. · Zbl 1268.81046
[14] Ross Duncan and Simon Perdrix. Rewriting measurement-based quantum computations with generalised flow.\it Lecture Notes in Computer Science, 6199:285-296, 2010.doi: 10.1007/978-3-642-14162-1_24. · Zbl 1288.68069
[15] Ross Duncan and Simon Perdrix. Pivoting makes the zx-calculus complete for real stabil izers. \it Electronic Proceedings in Theoretical Computer Science, 2013. doi:10.4204/EPTCS. 171.5.
[16] Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. A complete axiomatisation of the zx-calculus for clifford+ t quantum mechanics. \it arXiv preprint arXiv:1705.11151, 2017.
[17] Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. Y-calculus: A language for real matrices derived from the zx-calculus. In \it Conference on Quantum Physics and Logics \it (QPL’17), 2017.
[18] Simon Perdrix and Quanlong Wang. Supplementarity is necessary for quantum diagram reasoning. In \it 41st International Symposium on Mathematical Foundations of Computer Sci- \it ence (MFCS 2016), volume 58 of \it Leibniz International Proceedings in Informatics (LIPIcs), pages 76:1-76:14, Krakow, Poland, August 2016. doi:10.4230/LIPIcs.MFCS.2016.76. · Zbl 1398.81015
[19] Christian Schröder de Witt and Vladimir Zamdzhiev. The zx-calculus is incomplete for quantum mechanics. \it Electronic Proceedings in Theoretical Computer Science, 2014. doi: 10.4204/EPTCS.172.20.
[20] Peter Selinger.Finite dimensional hilbert spaces are complete for dagger compact closed categories. \it Logical Methods in Computer Science, 8(4):1-12, 2012. doi:10.2168/ LMCS-8(3:06)2012. · Zbl 1250.18007
[21] Peter Selinger. Quantum circuits of \it t-depth one. \it Phys. Rev. A, 87:042302, Apr 2013. doi:10.1103/PhysRevA.87.042302.
[22] Robert Spekkens. Evidence for the epistemic view of quantum states: A toy theory. \it Phys. \it Rev. A, 75:032110, Mar 2007. doi:10.1103/PhysRevA.75.032110.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.