Document Open Access Logo

Quantum Relaxations of CSP and Structure Isomorphism

Author Amin Karamlou

PDF

File

Thumbnail PDF
PDF
LIPIcs.MFCS.2025.61.pdf
  • Filesize: 0.91 MB
  • 18 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum information theory
  • Theory of computation → Categorical semantics
Keywords
  • CSP
  • graph isomorphism
  • quantum information
  • non-local game
  • quantum graph homomorphism
  • monad

Metrics

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

Abstract

We investigate quantum relaxations of two key decision problems in computer science: the constraint satisfaction problem (CSP) and the structure isomorphism problem. CSP asks whether a homomorphism exists between two relational structures, while structure isomorphism seeks an isomorphism between them. In recent years, it has become increasingly apparent that many special cases of CSP can be reformulated in terms of the existence of perfect classical strategies in non-local games, a key topic of study in quantum information theory. These games have allowed us to study quantum advantage in relation to many important decision problems, such as the k-colouring problem, and the problem of solving binary constraint systems. Abramsky et al. (2017) have shown that all of these games can be seen as special instances of a non-local CSP game. Moreover, they show that perfect quantum strategies in this CSP game can be viewed as Kleisli morphisms of a graded monad on the category of relational structures, which they dub the quantum monad. In this way, the quantum monad provides a categorical characterisation of quantum advantage for the non-local CSP game.
In this work we solidify and expand the results of Abramsky et al., answering several of their open questions. Firstly, we compare the definition of quantum graph homomorphisms arising from this work with an earlier definition of the concept due to Mančinska and Roberson and show that there are graphs which exhibit quantum advantage under one definition but not the other. Our second contribution is to extend the results of Abramsky et al. which only hold in the tensor product framework of quantum mechanics to the commuting operator framework. Next, we study a non-local structure isomorphism game, which generalises the well-studied graph isomorphism game. We show how the construction of the quantum monad can be refined to provide categorical semantics for quantum strategies in this game. This results in a category where morphisms coincide with quantum homomorphisms and isomorphisms coincide with quantum isomorphisms.

Cite As Get BibTex

Amin Karamlou. Quantum Relaxations of CSP and Structure Isomorphism. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 61:1-61:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.61

Author Details

Amin Karamlou
  • University College London, UK
  • University of Oxford, UK

Funding

I acknowledge funding from the EPSRC project Resources in Computation [EP/V040944/1].

Acknowledgements

I would like to thank Samson Abrasmky and Nihil Shah for many helpful discussions. In particular, the idea of using specification structures to extract a total category from partial categories was suggested by and worked out with Samson Abramsky. I would also like to thank Sam Staton and Rui Soares Barbosa for detailed feedback on an earlier draft of this work which has led to several improvements in the current version.

References

  1. Samson Abramsky, Rui Soares Barbosa, Nadish de Silva, and Octavio Zapata. The Quantum Monad on Relational Structures. 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 35:1-35:19, Dagstuhl, Germany, 2017. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.MFCS.2017.35.
  2. Samson Abramsky, Simon Gay, and Rajagopal Nagarajan. Specification Structures and propositions-as-types for concurrency, pages 5-40. Springer Berlin Heidelberg, Berlin, Heidelberg, 1996. URL: https://doi.org/10.1007/3-540-60915-6_2.
  3. Robert Atkey. Parameterised notions of computation. Journal of functional programming, 19(3-4):335-376, 2009. URL: https://doi.org/10.1017/S095679680900728X.
  4. Albert Atserias, Laura Mančinska, David E. Roberson, Robert Šámal, Simone Severini, and Antonios Varvitsiotis. Quantum and non-signalling graph isomorphisms. Journal of Combinatorial Theory, Series B, 136:289-328, 2019. URL: https://doi.org/10.1016/j.jctb.2018.11.002.
  5. László Babai. Graph isomorphism in quasipolynomial time. CoRR, abs/1512.03547, 2015. URL: https://arxiv.org/abs/1512.03547.
  6. John S Bell. On the einstein podolsky rosen paradox. Physics Physique Fizika, 1(3):195, 1964. Google Scholar
  7. Stefano Bistarelli, Ugo Montanari, Francesca Rossi, Thomas Schiex, Gérard Verfaillie, and Hélène Fargier. Semiring-based csps and valued csps: Frameworks, properties, and comparison. Constraints, 4:199-240, September 1999. URL: https://doi.org/10.1023/A:1026441215081.
  8. Adán Cabello, Matthias Kleinmann, and José R Portillo. Quantum state-independent contextuality requires 13 rays. Journal of Physics A: Mathematical and Theoretical, 49(38):38LT01, August 2016. URL: https://doi.org/10.1088/1751-8113/49/38/38lt01.
  9. R. Cleve, P. Hoyer, B. Toner, and J. Watrous. Consequences and limits of nonlocal strategies. In Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004., pages 236-249, 2004. URL: https://doi.org/10.1109/CCC.2004.1313847.
  10. M Droste. Handbook of Weighted Automata. Springer-Verlag, 2009. Google Scholar
  11. Soichiro Fujii, Shin-ya Katsumata, and Paul-André Mellies. Towards a formal theory of graded monads. In International Conference on Foundations of Software Science and Computation Structures, pages 513-530. Springer, 2016. URL: https://doi.org/10.1007/978-3-662-49630-5_30.
  12. Erich Grädel, Hayyan Helal, Matthias Naaf, and Richard Wilke. Zero-one laws and almost sure valuations of first-order logic in semiring semantics. arXiv preprint arXiv:2203.03425, 2022. URL: https://doi.org/10.48550/arXiv.2203.03425.
  13. Todd J Green, Grigoris Karvounarakis, and Val Tannen. Provenance semirings. In Proceedings of the twenty-sixth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems, pages 31-40, 2007. Google Scholar
  14. Teiko Heinosaari, Daniel Reitzner, and Peter Stano. Notes on joint measurability of quantum observables. Foundations of Physics, 38(12):1133-1147, November 2008. URL: https://doi.org/10.1007/s10701-008-9256-7.
  15. Chris Heunen, Tobias Fritz, and Manuel L. Reyes. Quantum theory realizes all joint measurability graphs. Phys. Rev. A, 89:032121, March 2014. URL: https://doi.org/10.1103/PhysRevA.89.032121.
  16. Charles Antony Richard Hoare. An axiomatic basis for computer programming. Communications of the ACM, 12(10):576-580, 1969. URL: https://doi.org/10.1145/363235.363259.
  17. Bart Jacobs. Introduction to coalgebra, volume 59. Cambridge University Press, 2017. Google Scholar
  18. Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen. Mip*=re, 2020. URL: https://arxiv.org/abs/2001.04383.
  19. Simon Kochen and E. P. Specker. The problem of hidden variables in quantum mechanics. Journal of Mathematics and Mechanics, 17(1):59-87, 1967. URL: http://www.jstor.org/stable/24902153.
  20. Werner Kuich and Arto Salomaa. Semirings, automata, languages, volume 5. Springer Science & Business Media, 2012. Google Scholar
  21. Olivier Lalonde. On the quantum chromatic numbers of small graphs, 2023. URL: https://arxiv.org/abs/2311.08194.
  22. Martino Lupini, Laura Mančinska, and David E. Roberson. Nonlocal games and quantum permutation groups. Journal of Functional Analysis, 279(5):108592, 2020. URL: https://doi.org/10.1016/j.jfa.2020.108592.
  23. Laura Mančinska and David E Roberson. Quantum homomorphisms. Journal of Combinatorial Theory, Series B, 118:228-267, 2016. URL: https://doi.org/10.1016/J.JCTB.2015.12.009.
  24. Laura Mančinska and David E Roberson. Oddities of quantum colorings. arXiv preprint arXiv:1801.03542, 2018. Google Scholar
  25. Laura Mančinska and David E Roberson. Quantum isomorphism is equivalent to equality of homomorphism counts from planar graphs. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 661-672. IEEE, 2020. Google Scholar
  26. Mehryar Mohri. Weighted automata algorithms. Handbook of weighted automata, pages 213-254, 2009. Google Scholar
  27. Lovro Mrkonjić. The model theory of semiring semantics, 2020. Google Scholar
  28. Benjamin Musto, David Reutter, and Dominic Verdon. A compositional approach to quantum functions. Journal of Mathematical Physics, 59(8):081706, 2018. Google Scholar
  29. Benjamin Musto, David Reutter, and Dominic Verdon. The morita theory of quantum graph isomorphisms. Communications in Mathematical Physics, 365(2):797-845, 2019. Google Scholar
  30. Prem Nigam Kar, David E. Roberson, Tim Seppelt, and Peter Zeman. NPA Hierarchy for Quantum Isomorphism and Homomorphism Indistinguishability. arXiv e-prints, page arXiv:2407.10635, July 2024. URL: https://doi.org/10.48550/arXiv.2407.10635.
  31. Carlos M. Ortiz and Vern I. Paulsen. Quantum graph homomorphisms via operator systems. Linear Algebra and its Applications, 497:23-43, 2016. URL: https://doi.org/10.1016/j.laa.2016.02.019.
  32. Vern I. Paulsen, Simone Severini, Daniel Stahlke, Ivan G. Todorov, and Andreas Winter. Estimating quantum chromatic numbers. Journal of Functional Analysis, 270(6):2188-2222, 2016. URL: https://doi.org/10.1016/j.jfa.2016.01.010.
  33. Vern I. Paulsen and Ivan G. Todorov. Quantum chromatic numbers via operator systems. The Quarterly Journal of Mathematics, 66(2):677-692, March 2015. URL: https://doi.org/10.1093/qmath/hav004.
  34. Volkher B Scholz and Reinhard F Werner. Tsirelson’s problem. arXiv preprint arXiv:0812.4305, 2008. Google Scholar
  35. Boris Tsirelson. Bell inequalities and operator algebras. This was posted by Boris Tsirelson as problem, 33, 2006. Google Scholar
  36. Roope Uola, Tobias Moroder, and Otfried Gühne. Joint measurability of generalized measurements implies classicality. Physical Review Letters, 113(16), October 2014. URL: https://doi.org/10.1103/physrevlett.113.160403.
  37. Sixia Yu and C. H. Oh. State-independent proof of kochen-specker theorem with 13 rays. Physical Review Letters, 108(3), January 2012. URL: https://doi.org/10.1103/physrevlett.108.030402.
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