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Optimizing Quantum Circuit Parameters via SDP

Author Eunou Lee

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Eunou Lee
  • Sunkyunkwan University, Seoul, South Korea

Acknowledgements

This work was supported by the Postdoctoral Research Program of Sungkyunkwan University (2021).

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Eunou Lee. Optimizing Quantum Circuit Parameters via SDP. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 48:1-48:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ISAAC.2022.48

Abstract

In recent years, parameterized quantum circuits have become a major tool to design quantum algorithms for optimization problems. The challenge in fully taking advantage of a given family of parameterized circuits lies in finding a good set of parameters in a non-convex landscape that can grow exponentially to the number of parameters. We introduce a new framework for optimizing parameterized quantum circuits: round SDP solutions to circuit parameters. Within this framework, we propose an algorithm that produces approximate solutions for a quantum optimization problem called Quantum Max Cut. The rounding algorithm runs in polynomial time to the number of parameters regardless of the underlying interaction graph. The resulting 0.562-approximation algorithm for generic instances of Quantum Max Cut improves on the previously known best algorithms by Anshu, Gosset, and Morenz with a ratio 0.531 and by Parekh and Thompson with a ratio 0.533.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Quantum algorithm
  • Optimization
  • Rounding algorithm
  • Quantum Circuit
  • Approximation

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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. Anurag Anshu, David Gosset, Karen J. Morenz Korol, and Mehdi Soleimanifar. Improved approximation algorithms for bounded-degree local hamiltonians. Phys. Rev. Lett., 127:250502, December 2021. URL: https://doi.org/10.1103/PhysRevLett.127.250502.
  3. Nikhil Bansal, Sergey Bravyi, and Barbara M. Terhal. Classical approximation schemes for the ground-state energy of quantum and classical Ising spin Hamiltonians on planar graphs. Quant. Inf. Comp. Vol. 9, No.8, p. 0701 (2009), 2007. arXiv:0705.1115v4. URL: http://arxiv.org/abs/0705.1115.
  4. Fernando G. S. L. Brandão and Aram W. Harrow. Product-state approximations to quantum states. Communications in Mathematical Physics, 342(1):47-80, February 2016. URL: https://doi.org/10.1007/s00220-016-2575-1.
  5. Sergey Bravyi, David Gosset, Robert König, and Kristan Temme. Approximation algorithms for quantum many-body problems. Journal of Mathematical Physics, 60(3):032203, 2019. URL: https://doi.org/10.1063/1.5085428.
  6. Marco Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, et al. Variational quantum algorithms. Nature Reviews Physics, 3(9):625-644, 2021. Google Scholar
  7. Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A quantum approximate optimization algorithm, 2014. URL: https://doi.org/10.48550/arXiv.1411.4028.
  8. Sevag Gharibian and Julia Kempe. Approximation algorithms for QMA-complete problems. SIAM Journal on Computing 41(4): 1028-1050, 2012, 2011. arXiv:1101.3884v1. URL: https://doi.org/10.1137/110842272.
  9. Sevag Gharibian and Julia Kempe. Hardness of approximation for quantum problems, pages 387-398. Springer Berlin Heidelberg, Berlin, Heidelberg, 2012. URL: https://doi.org/10.1007/978-3-642-31594-7_33.
  10. 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 fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.31.
  11. 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.
  12. Sean Hallgren, Eunou Lee, and Ojas Parekh. An approximation algorithm for the max-2-local hamiltonian problem. In Jaroslaw Byrka and Raghu Meka, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2020, August 17-19, 2020, Virtual Conference, volume 176 of LIPIcs, pages 59:1-59:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.59.
  13. 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.
  14. John Kallaugher and Ojas Parekh. The quantum and classical streaming complexity of quantum and classical max-cut. CoRR, abs/2206.00213, 2022. URL: https://doi.org/10.48550/arXiv.2206.00213.
  15. Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O’Donnell. Optimal inapproximability results for max‐cut and other 2‐variable csps? SIAM Journal on Computing, 37(1):319-357, 2007. URL: https://doi.org/10.1137/S0097539705447372.
  16. Karen J Morenz Korol, Kenny Choo, and Antonio Mezzacapo. Quantum approximation algorithms for many-body and electronic structure problems. arXiv preprint, 2021. URL: http://arxiv.org/abs/2111.08090.
  17. 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.
  18. 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.
  19. Ojas Parekh and Kevin Thompson. Beating random assignment for approximating quantum 2-local hamiltonian problems. In Petra Mutzel, Rasmus Pagh, and Grzegorz Herman, editors, 29th Annual European Symposium on Algorithms, ESA 2021, September 6-8, 2021, Lisbon, Portugal (Virtual Conference), volume 204 of LIPIcs, pages 74:1-74:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ESA.2021.74.
  20. 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.
  21. Stephen Piddock and Ashley Montanaro. The complexity of antiferromagnetic interactions and 2d lattices. Quantum Inf. Comput., 17(7&8):636-672, 2017. URL: https://doi.org/10.26421/QIC17.7-8-6.
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