Document Open Access Logo

Quantum Circuit Completeness: Extensions and Simplifications

Authors Alexandre Clément , Noé Delorme , Simon Perdrix , Renaud Vilmart

PDF
Thumbnail PDF

File

LIPIcs.CSL.2024.20.pdf
  • Filesize: 0.99 MB
  • 23 pages

Document Identifiers

Author Details

Alexandre Clément
  • Université Paris-Saclay, ENS Paris-Saclay, CNRS, Inria, LMF, 91190, Gif-sur-Yvette, France
Noé Delorme
  • Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France
Simon Perdrix
  • Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France
Renaud Vilmart
  • Université Paris-Saclay, ENS Paris-Saclay, CNRS, Inria, LMF, 91190, Gif-sur-Yvette, France

Funding

This work is supported by the Plan France 2030 through the PEPR integrated project EPiQ ANR-22-PETQ-0007 and the HQI initiative ANR-22-PNCQ-0002; it is also supported by the ANR project SoftQPro ANR-17-CE25-0009-02, by the STIC-AmSud project Qapla’ 21-STIC-10, and by the European projects NEASQC and HPCQS.

Cite AsGet BibTex

Alexandre Clément, Noé Delorme, Simon Perdrix, and Renaud Vilmart. Quantum Circuit Completeness: Extensions and Simplifications. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 20:1-20:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.CSL.2024.20

Abstract

Although quantum circuits have been ubiquitous for decades in quantum computing, the first complete equational theory for quantum circuits has only recently been introduced. Completeness guarantees that any true equation on quantum circuits can be derived from the equational theory. We improve this completeness result in two ways: (i) We simplify the equational theory by proving that several rules can be derived from the remaining ones. In particular, two out of the three most intricate rules are removed, the third one being slightly simplified. (ii) The complete equational theory can be extended to quantum circuits with ancillae or qubit discarding, to represent respectively quantum computations using an additional workspace, and hybrid quantum computations. We show that the remaining intricate rule can be greatly simplified in these more expressive settings, leading to equational theories where all equations act on a bounded number of qubits. The development of simple and complete equational theories for expressive quantum circuit models opens new avenues for reasoning about quantum circuits. It provides strong formal foundations for various compiling tasks such as circuit optimisation, hardware constraint satisfaction and verification.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Equational logic and rewriting
Keywords
  • Quantum Circuits
  • Completeness
  • Graphical Language

Metrics

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

Related Versions

References

  1. Matthew Amy, Jianxin Chen, and Neil J. Ross. A finite presentation of CNOT-dihedral operators. Electronic Proceedings in Theoretical Computer Science, 266:84-97, February 2018. URL: https://doi.org/10.4204/eptcs.266.5.
  2. Atul Singh Arora, Andrea Coladangelo, Matthew Coudron, Alexandru Gheorghiu, Uttam Singh, and Hendrik Waldner. Quantum depth in the random oracle model. In Barna Saha and Rocco A. Servedio, editors, Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 1111-1124. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585153.
  3. Miriam Backens, Aleks Kissinger, Hector Miller-Bakewell, John van de Wetering, and Sal Wolffs. Completeness of the ZH-calculus. Compositionality, 5, July 2023. URL: https://doi.org/10.32408/compositionality-5-5.
  4. 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, March 2021. URL: https://doi.org/10.22331/q-2021-03-25-421.
  5. Adriano Barenco, Charles H. Bennett, Richard Cleve, David P. DiVincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A. Smolin, and Harald Weinfurter. Elementary gates for quantum computation. Physical Review A, 52(5):3457-3467, November 1995. URL: https://doi.org/10.1103/physreva.52.3457.
  6. Xiaoning Bian and Peter Selinger. Generators and relations for 2-qubit Clifford+T operators. Proceedings of QPL'22, Electronic Proceedings in Theoretical Computer Science, 394:13-28, November 2023. URL: https://doi.org/10.4204/eptcs.394.2.
  7. Xiaoning Bian and Peter Selinger. Generators and relations for 3-qubit Clifford+CS operators. Proceedings QPL'23, Electronic Proceedings in Theoretical Computer Science, 384:114-126, August 2023. URL: https://doi.org/10.4204/eptcs.384.7.
  8. 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. URL: https://doi.org/10.4230/LIPIcs.MFCS.2022.24.
  9. Titouan Carette, Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. Completeness of graphical languages for mixed state quantum mechanics. ACM Transactions on Quantum Computing, 2(4), December 2021. URL: https://doi.org/10.1145/3464693.
  10. Alexandre Clément, Noé Delorme, and Simon Perdrix. Minimal equational theories for quantum circuits, 2023. URL: https://arxiv.org/abs/2311.07476.
  11. Alexandre Clément, Nicolas Heurtel, Shane Mansfield, Simon Perdrix, and Benoît Valiron. LO_v-Calculus: A Graphical Language for Linear Optical Quantum Circuits. 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 35:1-35:16, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.MFCS.2022.35.
  12. Alexandre Clément, Nicolas Heurtel, Shane Mansfield, Simon Perdrix, and Benoît Valiron. A complete equational theory for quantum circuits. In Logic in Computer Science (LICS), 2023. Google Scholar
  13. Alexandre Clément, Noé Delorme, Simon Perdrix, and Renaud Vilmart. Quantum circuit completeness: Extensions and simplifications, 2023. URL: https://arxiv.org/abs/2303.03117.
  14. Robin Cockett and Cole Comfort. The category TOF. In Peter Selinger and Giulio Chiribella, editors, Proceedings 15th International Conference on Quantum Physics and Logic, QPL 2018, volume 287 of EPTCS, pages 67-84, 2019. Google Scholar
  15. Robin Cockett, Cole Comfort, and Priyaa Srinivasan. The category CNOT. In Peter Selinger and Giulio Chiribella, editors, Proceedings 15th International Conference on Quantum Physics and Logic, QPL 2018, volume 287 of EPTCS, pages 258-293, 2019. URL: https://doi.org/10.4204/EPTCS.266.18.
  16. 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. Google Scholar
  17. Vincent Danos, Elham Kashefi, Prakash Panangaden, and Simon Perdrix. Extended measurement calculus. Semantic techniques in quantum computation, pages 235-310, 2009. Google Scholar
  18. Niel de Beaudrap, Xiaoning Bian, and Quanlong Wang. Fast and Effective Techniques for T-Count Reduction via Spider Nest Identities. In Steven T. Flammia, editor, 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020), volume 158 of Leibniz International Proceedings in Informatics (LIPIcs), pages 11:1-11:23, Dagstuhl, Germany, 2020. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.TQC.2020.11.
  19. D. Deutsch. Quantum computational networks. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 425(1868):73-90, 1989. Google Scholar
  20. Amar Hadzihasanovic. The algebra of entanglement and the geometry of composition. PhD thesis, University of Oxford, 2017. Google Scholar
  21. 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.
  22. Atsuya Hasegawa and François Le Gall. An optimal oracle separation of classical and quantum hybrid schemes. In Sang Won Bae and Heejin Park, editors, 33rd International Symposium on Algorithms and Computation, ISAAC 2022, December 19-21, 2022, Seoul, Korea, volume 248 of LIPIcs, pages 6:1-6:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2022.6.
  23. Mathieu Huot and Sam Staton. Universal properties in quantum theory. 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 213-223, 2019. URL: https://doi.org/10.4204/EPTCS.287.12.
  24. Toshinari Itoko, Rudy Raymond, Takashi Imamichi, and Atsushi Matsuo. Optimization of quantum circuit mapping using gate transformation and commutation. Integration, 70:43-50, 2020. Google Scholar
  25. Kazuo Iwama, Yahiko Kambayashi, and Shigeru Yamashita. Transformation rules for designing CNOT-based quantum circuits. In Proceedings of the 39th annual Design Automation Conference, pages 419-424, 2002. Google Scholar
  26. Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. A complete axiomatisation of the ZX-calculus for Clifford+T quantum mechanics. In Anuj Dawar and Erich Grädel, editors, Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018, Oxford, UK, July 09-12, 2018, pages 559-568. ACM, 2018. URL: https://doi.org/10.1145/3209108.3209131.
  27. Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. Diagrammatic reasoning beyond Clifford+T quantum mechanics. In Anuj Dawar and Erich Grädel, editors, Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018, Oxford, UK, July 09-12, 2018, pages 569-578. ACM, 2018. URL: https://doi.org/10.1145/3209108.3209139.
  28. Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. A generic normal form for ZX-diagrams and application to the rational angle completeness. In 34th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2019, Vancouver, BC, Canada, June 24-27, 2019, pages 1-10. IEEE, 2019. URL: https://doi.org/10.1109/LICS.2019.8785754.
  29. 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.
  30. Aleks Kissinger and John van de Wetering. PyZX, 2018. URL: https://github.com/Quantomatic/pyzx.
  31. 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.
  32. Stephen Lack. Composing PROPs. In Theory and Applications of Categories, volume 13(9), pages 147-163, 2004. URL: http://www.tac.mta.ca/tac/volumes/13/9/13-09abs.html.
  33. Justin Makary, Neil J. Ross, and Peter Selinger. Generators and relations for real stabilizer operators. In Chris Heunen and Miriam Backens, editors, Proceedings of the 18th International Conference on Quantum Physics and Logic, QPL 2021, volume 343 of EPTCS, pages 14-36, 2021. URL: https://doi.org/10.4204/EPTCS.343.2.
  34. Dmitri Maslov, Gerhard W Dueck, D Michael Miller, and Camille Negrevergne. Quantum circuit simplification and level compaction. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 27(3):436-444, 2008. Google Scholar
  35. Dmitri Maslov, Christina Young, D Michael Miller, and Gerhard W Dueck. Quantum circuit simplification using templates. In Design, Automation and Test in Europe, pages 1208-1213. IEEE, 2005. Google Scholar
  36. D Michael Miller, Dmitri Maslov, and Gerhard W Dueck. A transformation based algorithm for reversible logic synthesis. In Proceedings of the 40th annual Design Automation Conference, pages 318-323, 2003. Google Scholar
  37. Cristopher Moore and Martin Nilsson. Parallel quantum computation and quantum codes. SIAM journal on computing, 31(3):799-815, 2001. Google Scholar
  38. Yunseong Nam, Neil J Ross, Yuan Su, Andrew M Childs, and Dmitri Maslov. Automated optimization of large quantum circuits with continuous parameters. npj Quantum Information, 4(1):1-12, 2018. Google Scholar
  39. Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2002. Google Scholar
  40. C.C. Paige and M. Wei. History and generality of the CS decomposition. Linear Algebra and its Applications, 208-209:303-326, 1994. URL: https://doi.org/10.1016/0024-3795(94)90446-4.
  41. 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 LICS, pages 1-14, 2023. URL: https://doi.org/10.1109/LICS56636.2023.10175672.
  42. André Ranchin and Bob Coecke. Complete set of circuit equations for stabilizer quantum mechanics. Physical Review A, 90(1):012109, 2014. Google Scholar
  43. Robert Raussendorf and Hans J Briegel. A one-way quantum computer. Physical review letters, 86(22):5188, 2001. Google Scholar
  44. Vivek Shende, Stephen Bullock, and Igor Markov. Synthesis of quantum logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 25, 2006-01-31 00:01:00 2006. URL: https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=150894.
  45. Sam Staton. Algebraic effects, linearity, and quantum programming languages. In Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL '15, pages 395-406, New York, NY, USA, 2015. Association for Computing Machinery. URL: https://doi.org/10.1145/2676726.2676999.
  46. W Forrest Stinespring. Positive functions on c^*-algebras. Proceedings of the American Mathematical Society, 6(2):211-216, 1955. Google Scholar
  47. 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. URL: https://doi.org/10.1109/LICS.2019.8785765.
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 Imprint Privacy Contact