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Complete ZX-Calculi for the Stabiliser Fragment in Odd Prime Dimensions

Authors Robert I. Booth , Titouan Carette



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Author Details

Robert I. Booth
  • University of Edinburgh, Edinburgh, United Kingdom
  • Sorbonne Université, CNRS, LIP6, 4 place Jussieu, F-75005 Paris, France
  • LORIA CNRS, Inria Mocqua, Université de Lorraine, F-54000 Nancy, France
Titouan Carette
  • Université Paris-Saclay, Inria, CNRS, LMF, 91190, Gif-sur-Yvette, France

Acknowledgements

We thank Cole Comfort and Simon Perdrix for enlightening discussions.

Cite AsGet BibTex

Robert I. Booth and Titouan Carette. Complete ZX-Calculi for the Stabiliser Fragment in Odd Prime Dimensions. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 24:1-24:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.MFCS.2022.24

Abstract

We introduce a family of ZX-calculi which axiomatise the stabiliser fragment of quantum theory in odd prime dimensions. These calculi recover many of the nice features of the qubit ZX-calculus which were lost in previous proposals for higher-dimensional systems. We then prove that these calculi are complete, i.e. provide a set of rewrite rules which can be used to prove any equality of stabiliser quantum operations. Adding a discard construction, we obtain a calculus complete for mixed state stabiliser quantum mechanics in odd prime dimensions, and this furthermore gives a complete axiomatisation for the related diagrammatic language for affine co-isotropic relations.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
Keywords
  • ZX-calculus
  • completeness
  • quantum
  • stabiliser
  • qudits

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References

  1. Samson Abramsky and Bob Coecke. Categorical quantum mechanics, 2008. URL: http://arxiv.org/abs/0808.1023.
  2. D. M. Appleby. Properties of the extended Clifford group with applications to SIC-POVMs and MUBs, 2009. URL: http://arxiv.org/abs/0909.5233.
  3. Miriam Backens. The ZX-calculus is complete for stabilizer quantum mechanics. New Journal of Physics, 16(9):093021, 2014. URL: https://doi.org/10.1088/1367-2630/16/9/093021.
  4. Miriam Backens. Making the stabilizer ZX-calculus complete for scalars. Electronic Proceedings in Theoretical Computer Science, 195:17-32, 2015. URL: https://doi.org/10.4204/EPTCS.195.2.
  5. Miriam Backens, Hector Miller-Bakewell, Giovanni de Felice, Leo Lobski, and John van de Wetering. There and back again: A circuit extraction tale. Quantum, 5:421, 2021. URL: https://doi.org/10.22331/q-2021-03-25-421.
  6. Miriam Backens, Simon Perdrix, and Quanlong Wang. A Simplified Stabilizer ZX-calculus. Electronic Proceedings in Theoretical Computer Science, 236:1-20, 2017. URL: https://doi.org/10.4204/EPTCS.236.1.
  7. John C. Baez, Brandon Coya, and Franciscus Rebro. Props in Network Theory. Theory and Application of Categories, 33(25):727-783, 2018. URL: http://arxiv.org/abs/1707.08321.
  8. John C. Baez and Brendan Fong. A Compositional Framework for Passive Linear Networks. Theory and Application of Categories, 33(38):1158-1222, 2018. URL: http://arxiv.org/abs/1504.05625.
  9. Xiaoning Bian and Quanlong Wang. Graphical Calculus for Qutrit Systems. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, E98.A(1):391-399, 2015. URL: https://doi.org/10.1587/transfun.E98.A.391.
  10. Filippo Bonchi, Fabio Gadducci, Aleks Kissinger, Pawel Sobocinski, and Fabio Zanasi. String Diagram Rewrite Theory II: Rewriting with Symmetric Monoidal Structure, 2021. URL: https://doi.org/10.48550/arXiv.2104.14686.
  11. Filippo Bonchi, Fabio Gadducci, Aleks Kissinger, Paweł Sobocinski, and Fabio Zanasi. String Diagram Rewrite Theory I: Rewriting with Frobenius Structure. Journal of the ACM (JACM), 2022. URL: https://doi.org/10.1145/3502719.
  12. Filippo Bonchi, Fabio Gadducci, Aleks Kissinger, Paweł Sobociński, and Fabio Zanasi. String Diagram Rewrite Theory III: Confluence with and without Frobenius, 2022. URL: https://doi.org/10.48550/arXiv.2109.06049.
  13. Filippo Bonchi, Pawel Sobocinski, and Fabio Zanasi. Interacting Hopf Algebras. Journal of Pure and Applied Algebra, 221(1):144-184, 2017. URL: https://doi.org/10.1016/j.jpaa.2016.06.002.
  14. Robert I. Booth and Titouan Carette. Complete ZX-calculi for the stabiliser fragment in odd prime dimensions, 2022. URL: http://arxiv.org/abs/2204.12531.
  15. Titouan Carette. When Only Topology Matters, 2021. URL: http://arxiv.org/abs/2102.03178.
  16. Titouan Carette. Wielding the ZX-calculus, Flexsymmetry, Mixed States, and Scalable Notations. Theses, Université de Lorraine, 2021. URL: https://hal.archives-ouvertes.fr/tel-03468027.
  17. Titouan Carette, Dominic Horsman, and Simon Perdrix. SZX-calculus: Scalable Graphical Quantum Reasoning, 2019. URL: http://arxiv.org/abs/1905.00041.
  18. Titouan Carette, Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. Completeness of graphical languages for mixed states quantum mechanics. In Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, and Stefano Leonardi, editors, 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece, volume 132 of LIPIcs, pages 108:1-108:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.108.
  19. Titouan Carette and Simon Perdrix. Colored props for large scale graphical reasoning, 2020. URL: http://arxiv.org/abs/2007.03564.
  20. Nicholas Chancellor, Aleks Kissinger, Joschka Roffe, Stefan Zohren, and Dominic Horsman. Graphical Structures for Design and Verification of Quantum Error Correction, 2018. URL: http://arxiv.org/abs/1611.08012.
  21. Sean Clark. Valence bond solid formalism for d-level one-way quantum computation. Journal of Physics A: Mathematical and General, 39(11):2701-2721, 2006. URL: https://doi.org/10.1088/0305-4470/39/11/010.
  22. Bob Coecke and Ross Duncan. Interacting Quantum Observables: Categorical Algebra and Diagrammatics. New Journal of Physics, 13(4):043016, 2011. URL: https://doi.org/10.1088/1367-2630/13/4/043016.
  23. Bob Coecke and Aleks Kissinger. Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning. Cambridge University Press, 2017. URL: https://doi.org/10.1017/9781316219317.
  24. Cole Comfort. Relational semantics for quantum algorithms. Google Scholar
  25. Cole Comfort. A symplectic setting for mixed stabiliser circuits, 2021. Google Scholar
  26. Cole Comfort and Aleks Kissinger. A Graphical Calculus for Lagrangian Relations, 2021. URL: http://arxiv.org/abs/2105.06244.
  27. Niel de Beaudrap. A linearized stabilizer formalism for systems of finite dimension, 2012. URL: http://arxiv.org/abs/1102.3354.
  28. Niel de Beaudrap, Xiaoning Bian, and Quanlong Wang. Fast and effective techniques for T-count reduction via spider nest identities, 2020. URL: http://arxiv.org/abs/2004.05164.
  29. Niel de Beaudrap and Dominic Horsman. The ZX calculus is a language for surface code lattice surgery, 2017. URL: http://arxiv.org/abs/1704.08670.
  30. Ross Duncan, Aleks Kissinger, Simon Perdrix, and John van de Wetering. Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus. Quantum, 4:279, 2020. URL: https://doi.org/10.22331/q-2020-06-04-279.
  31. Ross Duncan and Maxime Lucas. Verifying the Steane code with Quantomatic. Electronic Proceedings in Theoretical Computer Science, 171:33-49, 2014. URL: https://doi.org/10.4204/EPTCS.171.4.
  32. Ross Duncan and Simon Perdrix. Rewriting Measurement-Based Quantum Computations with Generalised Flow. In Samson Abramsky, Cyril Gavoille, Claude Kirchner, Friedhelm Meyer auf der Heide, and Paul G. Spirakis, editors, Automata, Languages and Programming, volume 6199, pages 285-296. Springer Berlin Heidelberg, 2010. URL: https://doi.org/10.1007/978-3-642-14162-1_24.
  33. Liam Garvie and Ross Duncan. Verifying the Smallest Interesting Colour Code with Quantomatic. Electronic Proceedings in Theoretical Computer Science, 266:147-163, 2018. URL: https://doi.org/10.4204/EPTCS.266.10.
  34. Vlad Gheorghiu. Standard form of qudit stabilizer groups. Physics Letters A, page 5, 2014. Google Scholar
  35. Daniel Gottesman. Fault-Tolerant Quantum Computation with Higher-Dimensional Systems. Chaos, Solitons & Fractals, 10(10):1749-1758, 1999. URL: https://doi.org/10.1016/S0960-0779(98)00218-5.
  36. Ladina Hausmann, Nuriya Nurgalieva, and Lídia del Rio. A consolidating review of Spekkens' toy theory. URL: http://arxiv.org/abs/2105.03277.
  37. Mark Howard and Jiri Vala. Qudit versions of the qubit "pi-over-eight" gate. Physical Review A, 86(2):022316, 2012. URL: https://doi.org/10.1103/PhysRevA.86.022316.
  38. Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics, 2017. URL: http://arxiv.org/abs/1705.11151.
  39. Aleks Kissinger and John van de Wetering. Reducing T-count with the ZX-calculus, 2020. URL: http://arxiv.org/abs/1903.10477.
  40. Aleks Kissinger, John van de Wetering, and Renaud Vilmart. Classical simulation of quantum circuits with partial and graphical stabiliser decompositions, 2022. URL: http://arxiv.org/abs/2202.09202.
  41. Markus Nenhauser. An Explicit Construction of the Metaplectic Representation over a Finite Field. Journal of Lie Theory, 12(15), 2002. Google Scholar
  42. Kang Feng Ng and Quanlong Wang. A universal completion of the ZX-calculus, 2017. URL: http://arxiv.org/abs/1706.09877.
  43. André Ranchin. Depicting qudit quantum mechanics and mutually unbiased qudit theories. Electronic Proceedings in Theoretical Computer Science, 172:68-91, 2014. URL: https://doi.org/10.4204/EPTCS.172.6.
  44. Peter Selinger. Dagger compact closed categories and completely positive maps: (extended abstract). Electron. Notes Theor. Comput. Sci., 170:139-163, 2007. URL: https://doi.org/10.1016/j.entcs.2006.12.018.
  45. Alex Townsend-Teague and Konstantinos Meichanetzidis. Classifying Complexity with the ZX-Calculus: Jones Polynomials and Potts Partition Functions, 2021. URL: http://arxiv.org/abs/2103.06914.
  46. John van de Wetering. ZX-calculus for the working quantum computer scientist, 2020. URL: http://arxiv.org/abs/2012.13966.
  47. Renaud Vilmart. A Near-Optimal Axiomatisation of ZX-Calculus for Pure Qubit Quantum Mechanics, 2018. URL: http://arxiv.org/abs/1812.09114.
  48. Quanlong Wang. Qutrit ZX-calculus is Complete for Stabilizer Quantum Mechanics. Electronic Proceedings in Theoretical Computer Science, 266:58-70, 2018. URL: https://doi.org/10.4204/EPTCS.266.3.
  49. Quanlong Wang. A non-anyonic qudit ZW-calculus, 2021. URL: http://arxiv.org/abs/2109.11285.
  50. Quanlong Wang. Qufinite ZX-calculus: A unified framework of qudit ZX-calculi, 2021. URL: http://arxiv.org/abs/2104.06429.
  51. Quanlong Wang and Xiaoning Bian. Qutrit Dichromatic Calculus and Its Universality. Electronic Proceedings in Theoretical Computer Science, 172:92-101, 2014. URL: https://doi.org/10.4204/EPTCS.172.7.
  52. Yuchen Wang, Zixuan Hu, Barry C. Sanders, and Sabre Kais. Qudits and High-Dimensional Quantum Computing. Frontiers in Physics, 8, 2020. URL: https://doi.org/10.3389/fphy.2020.589504.
  53. Fabio Zanasi. Interacting Hopf Algebras: The Theory of Linear Systems. phdthesis, Ecole Normale Superieure de Lyon, 2018. URL: http://arxiv.org/abs/1805.03032.
  54. D. L. Zhou, B. Zeng, Z. Xu, and C. P. Sun. Quantum computation based on d-level cluster states. Physical Review A, 68(6):062303, 2003. URL: https://doi.org/10.1103/PhysRevA.68.062303.
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