Document Open Access Logo

Simple Qudit ZX and ZH Calculi, via Integrals

Authors Niel de Beaudrap , Richard D. P. East

PDF

File

Thumbnail PDF
PDF
LIPIcs.MFCS.2024.20.pdf
  • Filesize: 2.97 MB
  • 20 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
Keywords
  • ZX-calculus
  • ZH-calculus
  • qudits
  • string diagrams
  • discrete integrals

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

The ZX calculus and ZH calculus use diagrams to denote and compute properties of quantum operations, using "rewrite rules" to transform between diagrams which denote the same operator through a functorial semantic map. Different semantic maps give rise to different rewrite systems, which may prove more convenient for different purposes. Using discrete measures, we describe semantic maps for ZX and ZH diagrams, well-suited to analyse unitary circuits and measurements on qudits of any fixed dimension D > 1 as a single "ZXH-calculus". We demonstrate rewrite rules for the "stabiliser fragment" of the ZX calculus and a "multicharacter fragment" of the ZH calculus.

Cite As Get BibTex

Niel de Beaudrap and Richard D. P. East. Simple Qudit ZX and ZH Calculi, via Integrals. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.20

Author Details

Niel de Beaudrap
  • University of Sussex, Brighton, UK
Richard D. P. East
  • Haiqu

Acknowledgements

Thanks to Patrick Roy, Titouan Carette, John van de Wetering, and Robert Booth for technical discussions, and to the anonymous referees for suggestions on presentation.

References

  1. Dorit Aharonov. A simple proof that Toffoli and Hadamard are quantum universal, 2003. [arXiv:quant-ph/0301040]. URL: https://doi.org/10.48550/arXiv.quant-ph/0301040.
  2. Miriam Backens. Making the stabilizer ZX-calculus complete for scalars. In Chris Heunen, Peter Selinger, and Jamie Vicary, editors, Proceedings of the 12th International Workshop on Quantum Physics and Logic (QPL 2015), Electronic Proceedings in Theoretical Computer Science, pages 17-32. Open Publishing Association, november 2015. See also [arXiv:1507.03854]. URL: https://doi.org/10.4204/EPTCS.195.2.
  3. Miriam Backens and Aleks Kissinger. ZH: A complete graphical calculus for quantum computations involving classical non-linearity. Electronic Proceedings in Theoretical Computer Science, 287:23-42, January 2019. See also [arXiv:1805.02175]. URL: https://doi.org/10.4204/eptcs.287.2.
  4. Miriam Backens, Aleks Kissinger, Hector Miller-Bakewell, John van de Wetering, and Sal Wolffs. Completeness of the ZH-calculus. Compositionality, Volume 5 (2023), July 2023. See also [arXiv:2103.06610]. URL: https://doi.org/10.32408/compositionality-5-5.
  5. Miriam Backens, Simon Perdrix, and Quanlong Wang. A simplified stabilizer ZX-calculus. Electronic Proceedings in Theoretical Computer Science, 236:1-20, January 2017. See also [arXiv:1602.04744]. URL: https://doi.org/10.4204/eptcs.236.1.
  6. Robert I. Booth and Titouan Carette. Complete ZX-calculi for the stabiliser fragment in odd prime dimensions. In Stefan Szeider, Robert Ganian, and Alexandra Silva, editors, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022), volume 241 of Leibniz International Proceedings in Informatics (LIPIcs), pages 24:1-24:15, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. See also [arXiv:2204.12531]. URL: https://doi.org/10.4230/LIPIcs.MFCS.2022.24.
  7. Titouan Carette. Wielding the ZX-calculus, Flexsymmetry, Mixed States, and Scalable Notations. Theses, Université de Lorraine, November 2021. URL: https://hal.science/tel-03468027.
  8. Titouan Carette, Yohann D'Anello, and Simon Perdrix. Quantum algorithms and oracles with the scalable ZX-calculus. Electronic Proceedings in Theoretical Computer Science, 343:193-209, September 2021. See also [arXiv:2104.01043]. URL: https://doi.org/10.4204/eptcs.343.10.
  9. Titouan Carette, Dominic Horsman, and Simon Perdrix. SZX-Calculus: Scalable Graphical Quantum Reasoning. In Peter Rossmanith, Pinar Heggernes, and Joost-Pieter Katoen, editors, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019), volume 138 of Leibniz International Proceedings in Informatics (LIPIcs), pages 55:1-55:15, Dagstuhl, Germany, 2019. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. See also [arXiv:1905.00041]. URL: https://doi.org/10.4230/LIPIcs.MFCS.2019.55.
  10. Bob Coecke and Ross Duncan. Interacting quantum observables: categorical algebra and diagrammatics. New Journal of Physics, 13(4):043016, April 2011. See also [arXiv:0906.4725]. URL: https://doi.org/10.1088/1367-2630/13/4/043016.
  11. Bob Coecke and Aleks Kissinger. Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning. Cambridge University Press, 2017. Google Scholar
  12. Bob Coecke and Quanlong Wang. ZX-rules for 2-qubit Clifford+T quantum circuits. In International Conference on Reversible Computation, pages 144-161. Springer, 2018. See also [arXiv:1804.05356]. URL: https://doi.org/10.1007/978-3-319-99498-7_10.
  13. Niel De Beaudrap. A linearized stabilizer formalism for systems of finite dimension. Quantum Information & Computation, 13(1-2):73-115, january 2013. See also [arXiv:1102.3354]. Google Scholar
  14. Niel de Beaudrap. Well-tempered ZX and ZH calculi. Electronic Proceedings in Theoretical Computer Science, 340:13-45, September 2021. See also [arXiv:2006.02557]. URL: https://doi.org/10.4204/eptcs.340.2.
  15. Niel de Beaudrap and Richard D. P. East. Simple ZX and ZH calculi for arbitrary finite dimensions, via discrete integrals, 2023. [arXiv:2304.03310v2] - version 2. URL: https://doi.org/10.48550/arXiv.2304.03310.
  16. Niel de Beaudrap, Aleks Kissinger, and Konstantinos Meichanetzidis. Tensor network rewriting strategies for satisfiability and counting. In Benoît Valiron, Shane Mansfield, Pablo Arrighi, and Prakash Panangaden, editors, Proceedings 17th International Conference on Quantum Physics and Logic, Paris, France, June 2 - 6, 2020, volume 340 of Electronic Proceedings in Theoretical Computer Science, pages 46-59. Open Publishing Association, 2021. See also [arXiv:2004.06455]. URL: https://doi.org/10.4204/EPTCS.340.3.
  17. Giovanni de Felice and Bob Coecke. Quantum linear optics via string diagrams. In Stefano Gogioso and Matty Hoban, editors, Proceedings 19th International Conference on Quantum Physics and Logic, Wolfson College, Oxford, UK, 27 June - 1 July 2022, volume 394 of Electronic Proceedings in Theoretical Computer Science, pages 83-100. Open Publishing Association, 2023. See also [arXiv:2204.12985]. URL: https://doi.org/10.4204/EPTCS.394.6.
  18. Richard D. P. East, Pierre Martin-Dussaud, and John van de Wetering. Spin-networks in the ZX-calculus, 2021. [arXiv:2111.03114]. URL: https://doi.org/10.48550/arXiv.2111.03114.
  19. Richard D.P. East, John van de Wetering, Nicholas Chancellor, and Adolfo G. Grushin. AKLT-states as ZX-diagrams: Diagrammatic reasoning for quantum states. PRX Quantum, 3:010302, january 2022. See also [arXiv:2012.01219. URL: https://doi.org/10.1103/PRXQuantum.3.010302.
  20. Xiaoyan Gong and Quanlong Wang. Equivalence of local complementation and Euler decomposition in the qutrit ZX-calculus, 2017. [arXiv:1704.05955]. Google Scholar
  21. Emmanuel Jeandel, Simon Perdrix, and Margarita Veshchezerova. Addition and differentiation of ZX-diagrams. In Amy P. Felty, editor, 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022), volume 228 of Leibniz International Proceedings in Informatics (LIPIcs), pages 13:1-13:19, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. [arXiv:2202.11386]. URL: https://doi.org/10.4230/LIPIcs.FSCD.2022.13.
  22. Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. A complete axiomatisation of the ZX-calculus for Clifford+T quantum mechanics. In 2018 33rd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), LICS '18, pages 559-568, New York, NY, USA, 2018. Association for Computing Machinery. See also [arXiv:1705.11151]. URL: https://doi.org/10.1145/3209108.3209131.
  23. Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. Completeness of the ZX-calculus. Logical Methods in Computer Science, June 2020. See also [arXiv:1903.06035]. URL: https://doi.org/10.23638/LMCS-16(2:11)2020.
  24. Emmanuel Jeandel, Simon Perdrix, Renaud Vilmart, and Quanlong Wang. ZX-calculus: Cyclotomic supplementarity and incompleteness for Clifford+T quantum mechanics. In Kim G. Larsen, Hans L. Bodlaender, and Jean-Francois Raskin, editors, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017), volume 83 of Leibniz International Proceedings in Informatics (LIPIcs), pages 11:1-11:13, Dagstuhl, Germany, 2017. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. See also [arXiv:1702.01945]. URL: https://doi.org/10.4230/LIPIcs.MFCS.2017.11.
  25. Stach Kuijpers, John van de Wetering, and Aleks Kissinger. Graphical Fourier theory and the cost of quantum addition, 2019. [arXiv:1904.07551]. Google Scholar
  26. Tuomas Laakkonen, Konstantinos Meichanetzidis, and John van de Wetering. A graphical #SAT algorithm for formulae with small clause density, 2022. [arXiv:2212.08048]. URL: https://doi.org/10.48550/arXiv.2212.08048.
  27. Shahn Majid. Quantum and braided ZX calculus. Journal of Physics A: Mathematical and Theoretical, 55(25):254007, june 2022. See also [arXiv:2103.07264]. URL: https://doi.org/10.1088/1751-8121/ac631f.
  28. Kang Feng Ng and Quanlong Wang. A universal completion of the ZX-calculus, 2017. [arXiv:1706.09877]. URL: https://doi.org/10.48550/arXiv.1706.09877.
  29. Kang Feng Ng and Quanlong Wang. Completeness of the ZX-calculus for pure qubit Clifford+T quantum mechanics, January 2018. URL: https://doi.org/10.48550/arXiv.1801.07993.
  30. Simon Perdrix and Quanlong Wang. Supplementarity is necessary for quantum diagram reasoning. In Piotr Faliszewski, Anca Muscholl, and Rolf Niedermeier, editors, 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, Dagstuhl, Germany, 2016. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. See also [arXiv:1506.03055]. URL: https://doi.org/10.4230/LIPIcs.MFCS.2016.76.
  31. Boldizsár Poór, Robert I. Booth, Titouan Carette, John van de Wetering, and Lia Yeh. The qupit stabiliser ZX-travaganza: Simplified axioms, normal forms and graph-theoretic simplification. Electronic Proceedings in Theoretical Computer Science, 384:220-264, August 2023. See also [arXiv:2306.05204]. URL: https://doi.org/10.4204/eptcs.384.13.
  32. Boldizsár Poór, Quanlong Wang, Razin A. Shaikh, Lia Yeh, Richie Yeung, and Bob Coecke. Completeness for arbitrary finite dimensions of ZXW-calculus, a unifying calculus. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pages 1-14, 2023. See also [arXiv:2302.12135]. URL: https://doi.org/10.1109/LICS56636.2023.10175672.
  33. Patrick Roy. Qudit ZH-calculus. Master’s thesis, University of Oxford, 2022. Google Scholar
  34. Patrick Roy, John van de Wetering, and Lia Yeh. The qudit ZH-calculus: Generalised Toffoli+Hadamard and universality. Electronic Proceedings in Theoretical Computer Science, 384:142-170, August 2023. See also [arXiv:2307.10095]. URL: https://doi.org/10.4204/eptcs.384.9.
  35. Dirk Schlingemann. Cluster states, algorithms and graphs. Quantum Information & Computation, 4(4):287-324, july 2004. See also [arXiv:quant-ph/0305170]. Google Scholar
  36. Yaoyun Shi. Both Toffoli and controlled-NOT need little help to do universal quantum computing. Quantum Information & Computation, 3(1):84-92, january 2003. See also [arXiv:quant-ph/0205115]. Google Scholar
  37. Tobias Stollenwerk and Stuart Hadfield. Diagrammatic analysis for parameterized quantum circuits. In Stefano Gogioso and Matty Hoban, editors, Proceedings 19th International Conference on Quantum Physics and Logic, Wolfson College, Oxford, UK, 27 June - 1 July 2022, volume 394 of Electronic Proceedings in Theoretical Computer Science, pages 262-301. Open Publishing Association, 2023. See also [arXiv:2204.01307]. URL: https://doi.org/10.4204/EPTCS.394.15.
  38. Alexis Toumi, Richie Yeung, and Giovanni de Felice. Diagrammatic differentiation for quantum machine learning. In Chris Heunen and Miriam Backens, editors, Proceedings 18th International Conference on Quantum Physics and Logic, Gdansk, Poland, and online, 7-11 June 2021, volume 343 of Electronic Proceedings in Theoretical Computer Science, pages 132-144. Open Publishing Association, 2021. See also [arXiv:2103.07960]. URL: https://doi.org/10.4204/EPTCS.343.7.
  39. Alex Townsend-Teague and Konstantinos Meichanetzidis. Simplification strategies for the qutrit ZX-calculus, 2021. [arXiv:2103.06914]. URL: https://doi.org/10.48550/arXiv.2103.06914.
  40. John van de Wetering and Sal Wolffs. Completeness of the phase-free ZH-calculus, 2019. [arXiv:1904.07545]. URL: https://doi.org/10.48550/arXiv.1904.07545.
  41. John van de Wetering and Lia Yeh. Building qutrit diagonal gates from phase gadgets. Electronic Proceedings in Theoretical Computer Science, 394:46-65, November 2023. See also [arXiv:2204.13681]. URL: https://doi.org/10.4204/eptcs.394.4.
  42. Renaud Vilmart. A near-minimal axiomatisation of ZX-calculus for pure qubit quantum mechanics. In 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pages 1-10, 2019. See also [arXiv:1812.09114]. URL: https://doi.org/10.1109/LICS.2019.8785765.
  43. Renaud Vilmart. A ZX-calculus with triangles for Toffoli-Hadamard, Clifford+T, and beyond. Electronic Proceedings in Theoretical Computer Science, 287:313-344, January 2019. See also [arXiv:1804.03084]. URL: https://doi.org/10.4204/eptcs.287.18.
  44. Quanlong Wang. Qutrit ZX-calculus is complete for stabilizer quantum mechanics. Electronic Proceedings in Theoretical Computer Science, 266:58-70, February 2018. See also [arXiv:1803.00696]. URL: https://doi.org/10.4204/eptcs.266.3.
  45. Quanlong Wang. On completeness of algebraic ZX-calculus over arbitrary commutative rings and semirings, 2019. [arXiv:1912.01003]. URL: https://doi.org/10.48550/arXiv.1912.01003.
  46. Quanlong Wang. Algebraic complete axiomatisation of ZX-calculus with a normal form via elementary matrix operations, 2020. [arXiv:2007.13739]. URL: https://doi.org/10.48550/arXiv.2007.13739.
  47. Quanlong Wang. An algebraic axiomatisation of ZX-calculus. In Benoît Valiron, Shane Mansfield, Pablo Arrighi, and Prakash Panangaden, editors, Proceedings 17th International Conference on Quantum Physics and Logic, Paris, France, June 2 - 6, 2020, volume 340 of Electronic Proceedings in Theoretical Computer Science, pages 303-332. Open Publishing Association, 2021. See also [arXiv:1911.06752]. URL: https://doi.org/10.4204/EPTCS.340.16.
  48. Quanlong Wang. Qufinite ZX-calculus: a unified framework of qudit ZX-calculi, 2021. [arXiv:2104.06429]. URL: https://doi.org/10.48550/arXiv.2104.06429.
  49. Quanlong Wang, Richie Yeung, and Mark Koch. Differentiating and integrating ZX diagrams with applications to quantum machine learning, 2022. [arXiv:2201.13250]. URL: https://doi.org/10.48550/arXiv.2201.13250.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact