Document Open Access Logo

A Complete Dichotomy for Complex-Valued Holant^c

Author Miriam Backens

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2018.12.pdf
  • Filesize: 493 kB
  • 14 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Problems, reductions and completeness
Keywords
  • computational complexity
  • counting complexity
  • Holant problems
  • dichotomy
  • entanglement

Metrics

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

Abstract

Holant problems are a family of counting problems on graphs, parametrised by sets of complex-valued functions of Boolean inputs. Holant^c denotes a subfamily of those problems, where any function set considered must contain the two unary functions pinning inputs to values 0 or 1. The complexity classification of Holant problems usually takes the form of dichotomy theorems, showing that for any set of functions in the family, the problem is either #P-hard or it can be solved in polynomial time. Previous such results include a dichotomy for real-valued Holant^c and one for Holant^c with complex symmetric functions, i.e. functions which only depend on the Hamming weight of the input.
Here, we derive a dichotomy theorem for Holant^c with complex-valued, not necessarily symmetric functions. The tractable cases are the complex-valued generalisations of the tractable cases of the real-valued Holant^c dichotomy. The proof uses results from quantum information theory, particularly about entanglement. This full dichotomy for Holant^c answers a question that has been open for almost a decade.

Cite As Get BibTex

Miriam Backens. A Complete Dichotomy for Complex-Valued Holant^c. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 12:1-12:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.ICALP.2018.12

Author Details

Miriam Backens
  • Department of Computer Science, University of Oxford, UK

Funding

  • Backens, Miriam: The research leading to these results has received funding from EPSRC via grant EP/L021005/1 and from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) ERC grant agreement no. 334828. The paper reflects only the author’s views and not the views of the ERC or the European Commission. The European Union is not liable for any use that may be made of the information contained therein. No new data were created during this study. The majority of this research was done at the School of Mathematics, University of Bristol, UK.

References

  1. Miriam Backens. A complete dichotomy for complex-valued Holant^c. arXiv:1704.05798 [quant-ph], 2017. Google Scholar
  2. Miriam Backens. A New Holant Dichotomy Inspired by Quantum Computation. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017), volume 80 of Leibniz International Proceedings in Informatics (LIPIcs), pages 16:1-16:14, Dagstuhl, Germany, 2017. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2017.16.
  3. Miriam Backens. Number of superclasses of four-qubit entangled states under the inductive entanglement classification. Physical Review A, 95(2):022329, 2017. URL: http://dx.doi.org/10.1103/PhysRevA.95.022329.
  4. J. Cai, X. Chen, and P. Lu. Graph Homomorphisms with Complex Values: A Dichotomy Theorem. SIAM Journal on Computing, 42(3):924-1029, jan 2013. URL: http://dx.doi.org/10.1137/110840194.
  5. J. Cai, H. Guo, and T. Williams. A Complete Dichotomy Rises from the Capture of Vanishing Signatures. SIAM Journal on Computing, 45(5):1671-1728, 2016. URL: http://dx.doi.org/10.1137/15M1049798.
  6. J. Cai, P. Lu, and M. Xia. Dichotomy for Holant* Problems of Boolean Domain. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, Proceedings, pages 1714-1728. Society for Industrial and Applied Mathematics, jan 2011. URL: http://dx.doi.org/10.1137/1.9781611973082.132.
  7. Jin-Yi Cai and Vinay Choudhary. Valiant’s Holant Theorem and matchgate tensors. Theoretical Computer Science, 384(1):22-32, 2007. URL: http://dx.doi.org/10.1016/j.tcs.2007.05.015.
  8. Jin-Yi Cai and Zhiguo Fu. Holographic Algorithm with Matchgates is Universal for Planar #CSP over Boolean Domain. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, pages 842-855, New York, NY, USA, 2017. ACM. URL: http://dx.doi.org/10.1145/3055399.3055405.
  9. Jin-Yi Cai, Sangxia Huang, and Pinyan Lu. From Holant to #CSP and Back: Dichotomy for Holant^c Problems. Algorithmica, 64(3):511-533, mar 2012. URL: http://dx.doi.org/10.1007/s00453-012-9626-6.
  10. Jin-Yi Cai, Pinyan Lu, and Mingji Xia. The complexity of complex weighted Boolean #CSP. Journal of Computer and System Sciences, 80(1):217-236, feb 2014. First appeared as "Holant Problems and Counting CSP" in STOC '09. URL: http://dx.doi.org/10.1016/j.jcss.2013.07.003.
  11. Jin-Yi Cai, Pinyan Lu, and Mingji Xia. Dichotomy for Real Holantsuperscriptc Problems. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1802-1821. Society for Industrial and Applied Mathematics, 2018. URL: http://dx.doi.org/10.1137/1.9781611975031.118.
  12. W. Dür, G. Vidal, and J. I. Cirac. Three qubits can be entangled in two inequivalent ways. Physical Review A, 62(6):062314, 2000. URL: http://dx.doi.org/10.1103/PhysRevA.62.062314.
  13. Mariami Gachechiladze and Otfried Gühne. Completing the proof of “Generic quantum nonlocality”. Physics Letters A, 381(15):1281-1285, apr 2017. URL: http://dx.doi.org/10.1016/j.physleta.2016.10.001.
  14. Sangxia Huang and Pinyan Lu. A Dichotomy for Real Weighted Holant Problems. computational complexity, 25(1):255-304, mar 2016. URL: http://dx.doi.org/10.1007/s00037-015-0118-3.
  15. L. Lamata, J. León, D. Salgado, and E. Solano. Inductive classification of multipartite entanglement under stochastic local operations and classical communication. Physical Review A, 74(5):052336, nov 2006. URL: http://dx.doi.org/10.1103/PhysRevA.74.052336.
  16. L. Lamata, J. León, D. Salgado, and E. Solano. Inductive entanglement classification of four qubits under stochastic local operations and classical communication. Physical Review A, 75(2):022318, feb 2007. URL: http://dx.doi.org/10.1103/PhysRevA.75.022318.
  17. Dafa Li, Xiangrong Li, Hongtao Huang, and Xinxin Li. Simple criteria for the SLOCC classification. Physics Letters A, 359(5):428-437, dec 2006. URL: http://dx.doi.org/10.1016/j.physleta.2006.07.004.
  18. Jiabao Lin and Hanpin Wang. The Complexity of Holant Problems over Boolean Domain with Non-Negative Weights. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017), volume 80 of Leibniz International Proceedings in Informatics (LIPIcs), pages 29:1-29:14, Dagstuhl, Germany, 2017. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2017.29.
  19. Sandu Popescu and Daniel Rohrlich. Generic quantum nonlocality. Physics Letters A, 166(5–6):293-297, 1992. URL: http://dx.doi.org/10.1016/0375-9601(92)90711-T.
  20. L. Valiant. Holographic Algorithms. SIAM Journal on Computing, 37(5):1565-1594, jan 2008. URL: http://dx.doi.org/10.1137/070682575.
  21. F. Verstraete, J. Dehaene, B. De Moor, and H. Verschelde. Four qubits can be entangled in nine different ways. Physical Review A, 65(5):052112, apr 2002. URL: http://dx.doi.org/10.1103/PhysRevA.65.052112.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

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