Beyond Product State Approximations for a Quantum Analogue of Max Cut

Authors Anurag Anshu , David Gosset , Karen Morenz



PDF
Thumbnail PDF

File

LIPIcs.TQC.2020.7.pdf
  • Filesize: 466 kB
  • 15 pages

Document Identifiers

Author Details

Anurag Anshu
  • Institute for Quantum Computing, University of Waterloo, Canada
  • Department of Combinatorics and Optimization, University of Waterloo, Canada
  • Perimeter Institute for Theoretical Physics, Waterloo, Canada
David Gosset
  • Institute for Quantum Computing, University of Waterloo, Canada
  • Department of Combinatorics and Optimization, University of Waterloo, Canada
Karen Morenz
  • Institute for Quantum Computing, University of Waterloo, Canada
  • Department of Combinatorics and Optimization, University of Waterloo, Canada
  • Department of Chemistry, University of Toronto, Canada

Acknowledgements

We thank Sevag Gharibian, Hosho Katsura, Eunou Lee, and Ojas Parekh for comments and helpful discussions.

Cite AsGet BibTex

Anurag Anshu, David Gosset, and Karen Morenz. Beyond Product State Approximations for a Quantum Analogue of Max Cut. In 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 158, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.TQC.2020.7

Abstract

We consider a computational problem where the goal is to approximate the maximum eigenvalue of a two-local Hamiltonian that describes Heisenberg interactions between qubits located at the vertices of a graph. Previous work has shed light on this problem’s approximability by product states. For any instance of this problem the maximum energy attained by a product state is lower bounded by the Max Cut of the graph and upper bounded by the standard Goemans-Williamson semidefinite programming relaxation of it. Gharibian and Parekh described an efficient classical approximation algorithm for this problem which outputs a product state with energy at least 0.498 times the maximum eigenvalue in the worst case, and observe that there exist instances where the best product state has energy 1/2 of optimal. We investigate approximation algorithms with performance exceeding this limitation which are based on optimizing over tensor products of few-qubit states and shallow quantum circuits. We provide an efficient classical algorithm which achieves an approximation ratio of at least 0.53 in the worst case. We also show that for any instance defined by a 3 or 4-regular graph, there is an efficiently computable shallow quantum circuit that prepares a state with energy larger than the best product state (larger even than its semidefinite programming relaxation).

Subject Classification

ACM Subject Classification
  • Theory of computation
Keywords
  • Approximation algorithms
  • Quantum many-body systems

Metrics

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

References

  1. Hallgren, Lee, Parekh 2019. Announced in a contributed talk at QIP 2020 in Shenzhen, China. Google Scholar
  2. PW Anderson. Limits on the energy of the antiferromagnetic ground state. Physical Review, 83(6):1260, 1951. Google Scholar
  3. Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. Journal of the ACM (JACM), 45(3):501-555, 1998. Google Scholar
  4. 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. arXiv preprint arXiv:0705.1115, 2007. Google Scholar
  5. Fernando GSL Brandao and Aram W Harrow. Product-state approximations to quantum states. Communications in Mathematical Physics, 342(1):47-80, 2016. Google Scholar
  6. Sergey Bravyi, David Gosset, Robert König, and Kristan Temme. Approximation algorithms for quantum many-body problems. Journal of Mathematical Physics, 60(3):032203, 2019. Google Scholar
  7. Jop Briët, Fernando Mário de Oliveira Filho, and Frank Vallentin. The positive semidefinite grothendieck problem with rank constraint. In International Colloquium on Automata, Languages, and Programming, pages 31-42. Springer, 2010. Google Scholar
  8. Moses Charikar and Anthony Wirth. Maximizing quadratic programs: Extending grothendieck’s inequality. In 45th Annual IEEE Symposium on Foundations of Computer Science, pages 54-60. IEEE, 2004. Google Scholar
  9. Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028, 2014. Google Scholar
  10. Sevag Gharibian and Julia Kempe. Approximation algorithms for qma-complete problems. SIAM Journal on Computing, 41(4):1028-1050, 2012. Google Scholar
  11. Sevag Gharibian and Yi-Kai Liu. Approximation algorithms for the quantum heisenberg model. Private communication, 2020. Google Scholar
  12. Sevag Gharibian and Ojas Parekh. Almost optimal classical approximation algorithms for a quantum generalization of max-cut. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019), 145:31, 2019. Google Scholar
  13. Michel X Goemans and David P Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM (JACM), 42(6):1115-1145, 1995. Google Scholar
  14. Aram W Harrow and Ashley Montanaro. Extremal eigenvalues of local hamiltonians. Quantum, 1:6, 2017. Google Scholar
  15. Julia Kempe, Alexei Kitaev, and Oded Regev. The complexity of the local hamiltonian problem. In International Conference on Foundations of Software Technology and Theoretical Computer Science, pages 372-383. Springer, 2004. Google Scholar
  16. Alexei Yu Kitaev, Alexander Shen, and Mikhail N Vyalyi. Classical and quantum computation. Number 47. American Mathematical Soc., 2002. Google Scholar
  17. Elliott Lieb and Daniel Mattis. Ordering energy levels of interacting spin systems. Journal of Mathematical Physics, 3(4):749-751, 1962. Google Scholar
  18. Russell Merris. A note on laplacian graph eigenvalues. Linear algebra and its applications, 285(1-3):33-35, 1998. Google Scholar
  19. Stephen Piddock and Ashley Montanaro. The complexity of antiferromagnetic interactions and 2d lattices. arXiv preprint arXiv:1506.04014, 2015. Google Scholar