Nothing Special   »   [go: up one dir, main page]
Document Open Access Logo

Quantum Multiple-Valued Decision Diagrams in Graphical Calculi

Author Renaud Vilmart

PDF
Thumbnail PDF

File

LIPIcs.MFCS.2021.89.pdf
  • Filesize: 0.74 MB
  • 15 pages

Document Identifiers

Author Details

Renaud Vilmart
  • Université Paris-Saclay, CNRS, ENS Paris-Saclay, Inria, Laboratoire Méthodes Formelles, 91190, Gif-sur-Yvette, France

Cite As Get BibTex

Renaud Vilmart. Quantum Multiple-Valued Decision Diagrams in Graphical Calculi. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 89:1-89:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.MFCS.2021.89

Abstract

Graphical calculi such as the ZH-calculus are powerful tools in the study and analysis of quantum processes, with links to other models of quantum computation such as quantum circuits, measurement-based computing, etc.
A somewhat compact but systematic way to describe a quantum process is through the use of quantum multiple-valued decision diagrams (QMDDs), which have already been used for the synthesis of quantum circuits as well as for verification.
We show in this paper how to turn a QMDD into an equivalent ZH-diagram, and vice-versa, and show how reducing a QMDD translates in the ZH-Calculus, hence allowing tools from one formalism to be used into the other.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum information theory
  • Theory of computation → Equational logic and rewriting
Keywords
  • Quantum Computing
  • ZH-Calculus
  • Decision Diagrams

Metrics

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

Related Versions

References

  1. Matthew Amy. Towards large-scale functional verification of universal quantum circuits. In Peter Selinger and Giulio Chiribella, editors, rm Proceedings of the 15th International Conference on Quantum Physics and Logic, rm Halifax, Canada, 3-7th June 2018, volume 287 of Electronic Proceedings in Theoretical Computer Science, pages 1-21, 2019. URL: https://doi.org/10.4204/EPTCS.287.1.
  2. Miriam Backens and Aleks Kissinger. ZH: A complete graphical calculus for quantum computations involving classical non-linearity. In Peter Selinger and Giulio Chiribella, editors, rm Proceedings of the 15th International Conference on Quantum Physics and Logic, rm Halifax, Canada, 3-7th June 2018, volume 287 of Electronic Proceedings in Theoretical Computer Science, pages 23-42, 2019. URL: https://doi.org/10.4204/EPTCS.287.2.
  3. Miriam Backens, Aleks Kissinger, Hector Miller-Bakewell, John van de Wetering, and Sal Wolffs. Completeness of the ZH-calculus, 2021. Google Scholar
  4. Miriam Backens, Hector Miller-Bakewell, Giovanni de Felice, Leo Lobski, and John van de Wetering. There and back again: A circuit extraction tale. arXiv: Quantum Physics, 2020. URL: http://arxiv.org/abs/2003.01664.
  5. Lukas Burgholzer, Rudy Raymond, and Robert Wille. Verifying Results of the IBM Qiskit Quantum Circuit Compilation Flow, 2020. Google Scholar
  6. Lukas Burgholzer and Robert Wille. Advanced Equivalence Checking for Quantum Circuits. arXiv e-prints, 2020. URL: http://arxiv.org/abs/2004.08420.
  7. Nicholas Chancellor, Aleks Kissinger, Joschka Roffe, Stefan Zohren, and Dominic Horsman. Graphical structures for design and verification of quantum error correction, 2016. URL: http://arxiv.org/abs/1611.08012.
  8. Man-Duen Choi. Completely positive linear maps on complex matrices. Linear Algebra and its Applications, 10(3):285-290, 1975. URL: https://doi.org/10.1016/0024-3795(75)90075-0.
  9. Bob Coecke and Ross Duncan. Interacting quantum observables: Categorical algebra and diagrammatics. New Journal of Physics, 13(4):043016, April 2011. URL: https://doi.org/10.1088/1367-2630/13/4/043016.
  10. Bob Coecke and Aleks Kissinger. The compositional structure of multipartite quantum entanglement. In Automata, Languages and Programming, pages 297-308. Springer Berlin Heidelberg, 2010. URL: https://doi.org/10.1007/978-3-642-14162-1_25.
  11. Niel de Beaudrap, Xiaoning Bian, and Quanlong Wang. Fast and effective techniques for T-count reduction via spider nest identities, 2020. Google Scholar
  12. Niel de Beaudrap, Ross Duncan, Dominic Horsman, and Simon Perdrix. Pauli Fusion: a computational model to realise quantum transformations from ZX terms. In QPL'19 : International Conference on Quantum Physics and Logic, Los Angeles, United States, 2019. 12 pages + appendices. URL: https://hal.archives-ouvertes.fr/hal-02413388.
  13. Niel de Beaudrap and Dominic Horsman. The ZX-calculus is a language for surface code lattice surgery. CoRR, abs/1704.08670, 2017. URL: http://arxiv.org/abs/1704.08670.
  14. Ross Duncan and Maxime Lucas. Verifying the Steane code with Quantomatic. In Bob Coecke and Matty Hoban, editors, rm Proceedings of the 10th International Workshop on Quantum Physics and Logic, rm Castelldefels (Barcelona), Spain, 17th to 19th July 2013, volume 171 of Electronic Proceedings in Theoretical Computer Science, pages 33-49. Open Publishing Association, 2014. URL: https://doi.org/10.4204/EPTCS.171.4.
  15. Ross Duncan and Simon Perdrix. Rewriting measurement-based quantum computations with generalised flow. Lecture Notes in Computer Science, 6199:285-296, 2010. URL: https://doi.org/10.1007/978-3-642-14162-1_24.
  16. 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.
  17. Anne Hillebrand. Quantum Protocols involving Multiparticle Entanglement and their Representations. Master’s thesis, University of Oxford, 2011. URL: https://www.cs.ox.ac.uk/people/bob.coecke/Anne.pdf.
  18. Andrzej Jamiołkowski. Linear transformations which preserve trace and positive semidefiniteness of operators. Reports on Mathematical Physics, 3(4):275-278, 1972. URL: https://doi.org/10.1016/0034-4877(72)90011-0.
  19. Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. A complete axiomatisation of the ZX-calculus for Clifford+T quantum mechanics. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS '18, pages 559-568, New York, NY, USA, 2018. ACM. URL: https://doi.org/10.1145/3209108.3209131.
  20. Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. A Generic Normal Form for ZX-Diagrams and Application to the Rational Angle Completeness. In 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pages 1-10, June 2019. URL: https://doi.org/10.1109/LICS.2019.8785754.
  21. 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.
  22. Aleks Kissinger and John van de Wetering. Reducing the number of non-Clifford gates in quantum circuits. Phys. Rev. A, 102:022406, August 2020. URL: https://doi.org/10.1103/PhysRevA.102.022406.
  23. Louis Lemonnier, John van de Wetering, and Aleks Kissinger. Hypergraph simplification: Linking the path-sum approach to the ZH-calculus, 2020. URL: https://arxiv.org/abs/2003.13564.
  24. D. M. Miller and M. A. Thornton. QMDD: A Decision Diagram Structure for Reversible and Quantum Circuits. In 36th International Symposium on Multiple-Valued Logic (ISMVL'06), pages 30-30, 2006. URL: https://doi.org/10.1109/ISMVL.2006.35.
  25. P. Niemann, R. Wille, D. M. Miller, M. A. Thornton, and R. Drechsler. QMDDs: Efficient Quantum Function Representation and Manipulation. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 35(1):86-99, 2016. URL: https://doi.org/10.1109/TCAD.2015.2459034.
  26. Philipp Niemann, Robert Wille, and Rolf Drechsler. Advanced exact synthesis of Clifford+T circuits. Quantum Information Processing, 19(9):317, 2020. URL: https://doi.org/10.1007/s11128-020-02816-0.
  27. 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, June 2019. URL: https://doi.org/10.1109/LICS.2019.8785765.
  28. Renaud Vilmart. The Structure of Sum-Over-Paths, its Consequences, and Completeness for Clifford, 2020. URL: http://arxiv.org/abs/2003.05678.
  29. A. Zulehner, S. Hillmich, I. L. Markov, and R. Wille. Approximation of Quantum States Using Decision Diagrams. In 2020 25th Asia and South Pacific Design Automation Conference (ASP-DAC), pages 121-126, 2020. URL: https://doi.org/10.1109/ASP-DAC47756.2020.9045454.
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact