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Addition and Differentiation of ZX-Diagrams

Authors Emmanuel Jeandel , Simon Perdrix , Margarita Veshchezerova

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Author Details

Emmanuel Jeandel
  • LORIA, CNRS, Université de Lorraine, Inria Mocqua, Nancy, France
Simon Perdrix
  • LORIA, CNRS, Université de Lorraine, Inria Mocqua, Nancy, France
Margarita Veshchezerova
  • LORIA, CNRS, Université de Lorraine, Inria Mocqua, Nancy, France
  • EDF R&D, France

Funding

This work was supported in part by the CIFRE EDF/Loria Quantum Computing for Combinatorial Optimisation, the French National Research Agency (ANR) under the research projects SoftQPro ANR-17-CE25-0009-02 and VanQuTe ANR-17-CE24-0035, by the DGE of the French Ministry of Industry under the research project PIA-GDN/QuantEx P163746-484124, by the PEPR integrated project EPiQ, by the STIC-AmSud project Qapla’ 21-STIC-10, and by the European projects NEASQC (funded from the European Union’s Horizon 2020 research and innovation programme grant agreement No 951821) and HPCQS (European High-Performance Computing Joint Undertaking under grant agreement No 101018180).

Acknowledgements

The authors want to thank Bob Coecke, Harny Wang, and Richie Yeung for their availability and the fruitful discussions in the last few days prior to the submission.

Cite AsGet BibTex

Emmanuel Jeandel, Simon Perdrix, and Margarita Veshchezerova. Addition and Differentiation of ZX-Diagrams. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 13:1-13:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.FSCD.2022.13

Abstract

The ZX-calculus is a powerful framework for reasoning in quantum computing. It provides in particular a compact representation of matrices of interests. A peculiar property of the ZX-calculus is the absence of a formal sum allowing the linear combinations of arbitrary ZX-diagrams. The universality of the formalism guarantees however that for any two ZX-diagrams, the sum of their interpretations can be represented by a ZX-diagram. We introduce a general, inductive definition of the addition of ZX-diagrams, relying on the construction of controlled diagrams. Based on this addition technique, we provide an inductive differentiation of ZX-diagrams. Indeed, given a ZX-diagram with variables in the description of its angles, one can differentiate the diagram according to one of these variables. Differentiation is ubiquitous in quantum mechanics and quantum computing (e.g. for solving optimization problems). Technically, differentiation of ZX-diagrams is strongly related to summation as witnessed by the product rules. We also introduce an alternative, non inductive, differentiation technique rather based on the isolation of the variables. Finally, we apply our results to deduce a diagram for an Ising Hamiltonian.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Axiomatic semantics
Keywords
  • ZX calculus
  • Addition of ZX diagrams
  • Diagrammatic differentiation

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References

  1. Miriam Backens. The ZX-calculus is complete for stabilizer quantum mechanics. New Journal of Physics, 16(9):093021, September 2014. URL: https://doi.org/10.1088/1367-2630/16/9/093021.
  2. R. F. BLUTE, J. R. B. COCKETT, and R. A. G. SEELY. Differential categories. Mathematical Structures in Computer Science, 16(6):1049-1083, 2006. URL: https://doi.org/10.1017/S0960129506005676.
  3. M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles. Variational quantum algorithms. Nature Reviews Physics, 3(9):625-644, August 2021. URL: https://doi.org/10.1038/s42254-021-00348-9.
  4. Bob Coecke and Ross Duncan. Interacting quantum observables. In Luca Aceto, Ivan Damgård, Leslie Ann Goldberg, Magnús M. Halldórsson, Anna Ingólfsdóttir, and Igor Walukiewicz, editors, Automata, Languages and Programming, pages 298-310, Berlin, Heidelberg, 2008. Springer Berlin Heidelberg. Google Scholar
  5. Bob Coecke and Aleks Kissinger. Picturing Phases and Complementarity, pages 510-623. Cambridge University Press, 2017. URL: https://doi.org/10.1017/9781316219317.010.
  6. Ross Duncan and Simon Perdrix. Pivoting makes the ZX-calculus complete for real stabilizers. Electronic Proceedings in Theoretical Computer Science, 171:50-62, December 2014. URL: https://doi.org/10.4204/eptcs.171.5.
  7. Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A quantum approximate optimization algorithm, 2014. URL: http://arxiv.org/abs/1411.4028.
  8. Gian Giacomo Guerreschi and Mikhail Smelyanskiy. Practical optimization for hybrid quantum-classical algorithms. arXiv: Quantum Physics, 2017. Google Scholar
  9. Stuart Hadfield. On the Representation of Boolean and Real Functions as Hamiltonians for Quantum Computing. ACM Transactions on Quantum Computing, 2(4):1-21, December 2021. URL: https://doi.org/10.1145/3478519.
  10. Amar Hadzihasanovic. A diagrammatic axiomatisation for qubit entanglement. In 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, pages 573-584, 2015. URL: https://doi.org/10.1109/LICS.2015.59.
  11. Amar Hadzihasanovic, Kang Feng Ng, and Quanlong Wang. Two complete axiomatisations of pure-state qubit quantum computing. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS '18, pages 502-511, New York, NY, USA, 2018. ACM. URL: https://doi.org/10.1145/3209108.3209128.
  12. Emmanuel Jeandel, Simon Perdrix, and Margarita Veshchezerova. Addition and Differentiation of ZX-diagrams, 2022. URL: https://doi.org/10.48550/ARXIV.2202.11386.
  13. Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics. In The 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018, Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, pages 559-568, Oxford, United Kingdom, July 2018. URL: https://doi.org/10.1145/3209108.3209131.
  14. Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. Diagrammatic Reasoning beyond Clifford+T Quantum Mechanics. In The 33rd Annual Symposium on Logic in Computer Science, Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, pages 569-578, Oxford, United Kingdom, July 2018. URL: https://doi.org/10.1145/3209108.3209139.
  15. Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. A Generic Normal Form for ZX-Diagrams and Application to the Rational Angle Completeness. In Proceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS '19. IEEE Press, 2019. Google Scholar
  16. Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. Completeness of the ZX-Calculus. Logical Methods in Computer Science, Volume 16, Issue 2, June 2020. URL: https://doi.org/10.23638/LMCS-16(2:11)2020.
  17. 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 fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.MFCS.2017.11.
  18. Andrew Lucas. Ising formulations of many np problems. Frontiers in Physics, 2, 2014. URL: https://doi.org/10.3389/fphy.2014.00005.
  19. Kang Feng Ng and Quanlong Wang. Completeness of the ZX-calculus for Pure Qubit Clifford+T Quantum Mechanics, 2018. URL: http://arxiv.org/abs/1801.07993.
  20. Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Alán Aspuru-Guzik, and Jeremy L. O'Brien. A variational eigenvalue solver on a photonic quantum processor. Nature Communications, 5(1):4213, July 2014. URL: https://doi.org/10.1038/ncomms5213.
  21. John Preskill. Quantum Computing in the NISQ era and beyond. Quantum, 2:79, August 2018. URL: https://doi.org/10.22331/q-2018-08-06-79.
  22. M. H. Stone. On One-Parameter Unitary Groups in Hilbert Space. Annals of Mathematics, 33(3):643-648, 1932. URL: http://www.jstor.org/stable/1968538.
  23. Alexis Toumi, Richie Yeung, and Giovanni de Felice. Diagrammatic Differentiation for Quantum Machine Learning. arXiv e-prints, page arXiv:2103.07960, March 2021. URL: http://arxiv.org/abs/2103.07960.
  24. John van de Wetering. ZX-calculus for the working quantum computer scientist, 2020. URL: http://arxiv.org/abs/2012.13966.
  25. Renaud Vilmart. A near-optimal axiomatisation of ZX-calculus for pure qubit quantum mechanics. In Proceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), 2019. URL: http://arxiv.org/abs/arXiv:1812.09114.
  26. Quanlong Wang. An Algebraic Axiomatisation of ZX-calculus. Electronic Proceedings in Theoretical Computer Science, 340:303-332, September 2021. URL: https://doi.org/10.4204/eptcs.340.16.
  27. Quanlong Wang and Richie Yeung. Differentiating and Integrating ZX Diagrams, 2022. URL: http://arxiv.org/abs/2201.13250.
  28. Richie Yeung. Diagrammatic design and study of ansätze for quantum machine learning, 2020. URL: http://arxiv.org/abs/2011.11073.
  29. Chen Zhao and Xiao-Shan Gao. Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus. Quantum, 5:466, June 2021. URL: https://doi.org/10.22331/q-2021-06-04-466.
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