Document Open Access Logo

An Improved Quantum Max Cut Approximation via Maximum Matching

Authors Eunou Lee, Ojas Parekh

PDF
Thumbnail PDF

File

LIPIcs.ICALP.2024.105.pdf
  • Filesize: 0.62 MB
  • 11 pages

Document Identifiers

Author Details

Eunou Lee
  • Korea Institute for Advanced Study, Seoul, South Korea
Ojas Parekh
  • Sandia National Laboratories, Albuquerque, NM, USA

Funding

Eunou Lee was supported by Individual Grant No.CG093801 at Korea Institute for Advanced Study. This work is supported by a collaboration between the US DOE and other Agencies. Ojas Parekh was supported by the U.S. Department of Energy (DOE), Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator. Additional support is acknowledged from DOE, Office of Science, Office of Advanced Scientific Computing Research, Accelerated Research in Quantum Computing, Fundamental Algorithmic Research in Quantum Computing. This article has been authored by an employee of National Technology & Engineering Solutions of Sandia, LLC under Contract No. DE-NA0003525 with the U.S. Department of Energy (DOE). The employee owns all right, title and interest in and to the article and is solely responsible for its contents. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this article or allow others to do so, for United States Government purposes. The DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan https://www.energy.gov/downloads/doe-public-access-plan.

Acknowledgements

Eunou Lee appreciates helpful comments on the write-up by Andrus Giraldo.

Cite AsGet BibTex

Eunou Lee and Ojas Parekh. An Improved Quantum Max Cut Approximation via Maximum Matching. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 105:1-105:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.105

Abstract

Finding a high (or low) energy state of a given quantum Hamiltonian is a potential area to gain a provable and practical quantum advantage. A line of recent studies focuses on Quantum Max Cut, where one is asked to find a high energy state of a given antiferromagnetic Heisenberg Hamiltonian. In this work, we present a classical approximation algorithm for Quantum Max Cut that achieves an approximation ratio of 0.595, outperforming the previous best algorithms of Lee [Eunou Lee, 2022] (0.562, generic input graph) and King [King, 2023] (0.582, triangle-free input graph). The algorithm is based on finding the maximum weighted matching of an input graph and outputs a product of at most 2-qubit states, which is simpler than the fully entangled output states of the previous best algorithms.

Subject Classification

ACM Subject Classification
  • Theory of computation → Rounding techniques
Keywords
  • approximation
  • optimization
  • local Hamiltonian
  • rounding
  • SDP
  • matching

Metrics

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

References

  1. Anurag Anshu, David Gosset, and Karen Morenz. Beyond Product State Approximations for a Quantum Analogue of Max Cut. 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 7:1-7:15, Dagstuhl, Germany, 2020. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.TQC.2020.7.
  2. Jop Briët, Fernando Mário de Oliveira Filho, and Frank Vallentin. Grothendieck inequalities for semidefinite programs with rank constraint. Theory of Computing, 10(4):77-105, 2014. URL: https://doi.org/10.4086/toc.2014.v010a004.
  3. Jack Edmonds. Maximum matching and a polyhedron with 0,1-vertices. Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics, page 125, 1965. URL: https://api.semanticscholar.org/CorpusID:15379135.
  4. Sevag Gharibian, Yichen Huang, Zeph Landau, and Seung Woo Shin. Quantum hamiltonian complexity. Foundations and Trends® in Theoretical Computer Science, 10(3):159-282, 2015. URL: https://doi.org/10.1561/0400000066.
  5. Sevag Gharibian and Ojas Parekh. Almost Optimal Classical Approximation Algorithms for a Quantum Generalization of Max-Cut. In Dimitris Achlioptas and László A. Végh, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019), volume 145 of Leibniz International Proceedings in Informatics (LIPIcs), pages 31:1-31:17, Dagstuhl, Germany, 2019. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.31.
  6. Michel X. Goemans and David P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM, 42(6):1115-1145, November 1995. URL: https://doi.org/10.1145/227683.227684.
  7. Yeongwoo Hwang, Joe Neeman, Ojas Parekh, Kevin Thompson, and John Wright. Unique games hardness of quantum max-cut, and a vector-valued borell’s inequality, 2021. URL: https://doi.org/10.48550/arXiv.2111.01254.
  8. Robbie King. An improved approximation algorithm for quantum max-cut on triangle-free graphs. Quantum, 7:1180, November 2023. URL: https://doi.org/10.22331/q-2023-11-09-1180.
  9. Jean B. Lasserre. An explicit equivalent positive semidefinite program for nonlinear 0-1 programs. SIAM J. on Optimization, 12(3):756-769, March 2002. URL: https://doi.org/10.1137/S1052623400380079.
  10. Eunou Lee. Optimizing quantum circuit parameters via SDP. 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 48:1-48:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.ISAAC.2022.48.
  11. Miguel Navascues, Stefano Pironio, and Antonio Acín. Convergent hierarchy of semidefinite programs characterizing the set of quantum correlations. New Journal of Physics, 10, July 2008. URL: https://doi.org/10.1088/1367-2630/10/7/073013.
  12. Ojas Parekh and Kevin Thompson. Application of the Level-2 Quantum Lasserre Hierarchy in Quantum Approximation Algorithms. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021), volume 198 of Leibniz International Proceedings in Informatics (LIPIcs), pages 102:1-102:20, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.102.
  13. Ojas Parekh and Kevin Thompson. An optimal product-state approximation for 2-local quantum hamiltonians with positive terms. CoRR, abs/2206.08342, 2022. URL: https://doi.org/10.48550/arXiv.2206.08342.
  14. Jun Takahashi, Chaithanya Rayudu, Cunlu Zhou, Robbie King, Kevin Thompson, and Ojas Parekh. An su(2)-symmetric semidefinite programming hierarchy for quantum max cut, 2023. URL: https://arxiv.org/abs/2307.15688.
  15. Adam Bene Watts, Anirban Chowdhury, Aidan Epperly, J. William Helton, and Igor Klep. Relaxations and exact solutions to quantum max cut via the algebraic structure of swap operators, 2023. URL: https://arxiv.org/abs/2307.15661.
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