Document Open Access Logo

Quantum-Inspired Sublinear Algorithm for Solving Low-Rank Semidefinite Programming

Authors Nai-Hui Chia, Tongyang Li, Han-Hsuan Lin, Chunhao Wang

PDF
Thumbnail PDF

File

LIPIcs.MFCS.2020.23.pdf
  • Filesize: 0.6 MB
  • 15 pages

Document Identifiers

Author Details

Nai-Hui Chia
  • Department of Computer Science, University of Texas at Austin, TX, USA
Tongyang Li
  • Department of Computer Science, Institute for Advanced Computer Studies, and Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD, USA
Han-Hsuan Lin
  • Department of Computer Science, University of Texas at Austin, TX, USA
Chunhao Wang
  • Department of Computer Science, University of Texas at Austin, TX, USA

Funding

NHC, HHL, and CW were supported by Scott Aaronson’s Vannevar Bush Faculty Fellowship.
  • Li, Tongyang: TL was supported by IBM PhD Fellowship, QISE-NET Triplet Award (NSF DMR-1747426), and the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Quantum Algorithms Teams program.

Acknowledgements

We thank Scott Aaronson, Andr{á}s Gilyén, Ewin Tang, Ronald de Wolf, and anonymous reviewers for their detailed feedback on preliminary versions of this paper.

Cite AsGet BibTex

Nai-Hui Chia, Tongyang Li, Han-Hsuan Lin, and Chunhao Wang. Quantum-Inspired Sublinear Algorithm for Solving Low-Rank Semidefinite Programming. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.MFCS.2020.23

Abstract

Semidefinite programming (SDP) is a central topic in mathematical optimization with extensive studies on its efficient solvers. In this paper, we present a proof-of-principle sublinear-time algorithm for solving SDPs with low-rank constraints; specifically, given an SDP with m constraint matrices, each of dimension n and rank r, our algorithm can compute any entry and efficient descriptions of the spectral decomposition of the solution matrix. The algorithm runs in time O(m⋅poly(log n,r,1/ε)) given access to a sampling-based low-overhead data structure for the constraint matrices, where ε is the precision of the solution. In addition, we apply our algorithm to a quantum state learning task as an application. Technically, our approach aligns with 1) SDP solvers based on the matrix multiplicative weight (MMW) framework by Arora and Kale [TOC '12]; 2) sampling-based dequantizing framework pioneered by Tang [STOC '19]. In order to compute the matrix exponential required in the MMW framework, we introduce two new techniques that may be of independent interest: - Weighted sampling: assuming sampling access to each individual constraint matrix A₁,…,A_τ, we propose a procedure that gives a good approximation of A = A₁+⋯+A_τ. - Symmetric approximation: we propose a sampling procedure that gives the spectral decomposition of a low-rank Hermitian matrix A. To the best of our knowledge, this is the first sampling-based algorithm for spectral decomposition, as previous works only give singular values and vectors.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Spectral decomposition
  • Semi-definite programming
  • Quantum-inspired algorithm
  • Sublinear algorithm

Metrics

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

Related Versions

References

  1. Scott Aaronson. Shadow tomography of quantum states. In Proceedings of the 50th Annual ACM Symposium on Theory of Computing (STOC 2018), pages 325-338. ACM, 2018. URL: http://arxiv.org/abs/1711.01053.
  2. Zeyuan Allen-Zhu, Yin Tat Lee, and Lorenzo Orecchia. Using optimization to obtain a width-independent, parallel, simpler, and faster positive sdp solver. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2016), pages 1824-1831. Society for Industrial and Applied Mathematics, 2016. URL: http://arxiv.org/abs/1507.02259.
  3. Kurt M. Anstreicher. The volumetric barrier for semidefinite programming. Mathematics of Operations Research, 25(3):365-380, 2000. Google Scholar
  4. Joran van Apeldoorn and András Gilyén. Improvements in quantum SDP-solving with applications. In Proceedings of 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), volume 132 of Leibniz International Proceedings in Informatics, pages 99:1-99:15, 2019. URL: http://arxiv.org/abs/1804.05058.
  5. Joran van Apeldoorn, András Gilyén, Sander Gribling, and Ronald de Wolf. Quantum SDP-solvers: Better upper and lower bounds. In Proceedings of the 58th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2017). IEEE, 2017. URL: http://arxiv.org/abs/1705.01843.
  6. Sanjeev Arora, Elad Hazan, and Satyen Kale. The multiplicative weights update method: a meta-algorithm and applications. Theory of Computing, 8(1):121-164, 2012. Google Scholar
  7. Sanjeev Arora and Satyen Kale. A combinatorial, primal-dual approach to semidefinite programs. In Proceedings of the 39th Annual ACM Symposium on Theory of Computing (STOC 2007), pages 227-236. ACM, 2007. Google Scholar
  8. Juan Miguel Arrazola, Alain Delgado, Bhaskar Roy Bardhan, and Seth Lloyd. Quantum-inspired algorithms in practice, 2019. URL: http://arxiv.org/abs/1905.10415.
  9. Fernando G. S. L. Brandão, Amir Kalev, Tongyang Li, Cedric Yen-Yu Lin, Krysta M. Svore, and Xiaodi Wu. Quantum SDP solvers: Large speed-ups, optimality, and applications to quantum learning. In Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), volume 132, page 27. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2019. URL: http://arxiv.org/abs/1710.02581.
  10. Fernando G. S. L. Brandão and Krysta Svore. Quantum speed-ups for semidefinite programming. In Proceedings of the 58th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2017). IEEE, 2017. URL: http://arxiv.org/abs/1609.05537.
  11. Nai-Hui Chia, András Gilyén, Tongyang Li, Han-Hsuan Lin, Ewin Tang, and Chunhao Wang. Sampling-based sublinear low-rank matrix arithmetic framework for dequantizing quantum machine learning. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC 2020), page 387–400. ACM, 2020. URL: http://arxiv.org/abs/1910.06151.
  12. Nai-Hui Chia, Han-Hsuan Lin, and Chunhao Wang. Quantum-inspired sublinear classical algorithms for solving low-rank linear systems, 2018. URL: http://arxiv.org/abs/1811.04852.
  13. Michael B. Cohen, Yin Tat Lee, and Zhao Song. Solving linear programs in the current matrix multiplication time. In Proceedings of the 51st Annual ACM Symposium on Theory of Computing (STOC 2019), pages 938-942. ACM, 2019. URL: http://arxiv.org/abs/1810.07896.
  14. P. Drineas, R. Kannan, and M. Mahoney. Fast monte carlo algorithms for matrices i: Approximating matrix multiplication. SIAM Journal on Computing, 36(1):132-157, 2006. Google Scholar
  15. Alan Frieze, Ravi Kannan, and Santosh Vempala. Fast Monte-Carlo algorithms for finding low-rank approximations. Journal of the ACM, 51(6):1025-1041, 2004. Google Scholar
  16. Dan Garber and Elad Hazan. Approximating semidefinite programs in sublinear time. In Advances in Neural Information Processing Systems (NIPS 2011), pages 1080-1088, 2011. Google Scholar
  17. Dan Garber and Elad Hazan. Almost optimal sublinear time algorithm for semidefinite programming, 2012. URL: http://arxiv.org/abs/1208.5211.
  18. András Gilyén, Seth Lloyd, and Ewin Tang. Quantum-inspired low-rank stochastic regression with logarithmic dependence on the dimension, 2018. URL: http://arxiv.org/abs/1811.04909.
  19. Martin Grötschel, László Lovász, and Alexander Schrijver. The ellipsoid method and its consequences in combinatorial optimization. Combinatorica, 1(2):169-197, 1981. Google Scholar
  20. Gus Gutoski and Xiaodi Wu. Parallel approximation of min-max problems with applications to classical and quantum zero-sum games. In Proceedings of the 27th Annual IEEE Symposium on Computational Complexity (CCC 2012), pages 21-31. IEEE, 2012. Google Scholar
  21. Elad Hazan. Efficient algorithms for online convex optimization and their applications. PhD thesis, Princeton University, 2006. Google Scholar
  22. Rahul Jain and Penghui Yao. A parallel approximation algorithm for positive semidefinite programming. In Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2011), pages 463-471. IEEE, 2011. URL: http://arxiv.org/abs/1104.2502.
  23. Dhawal Jethwani, François Le Gall, and Sanjay K. Singh. Quantum-inspired classical algorithms for singular value transformation, 2019. URL: http://arxiv.org/abs/1910.05699.
  24. Haotian Jiang, Yin Tat Lee, Zhao Song, and Sam Chiu-wai Wong. An improved cutting plane method for convex optimization, convex-concave games and its applications. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC 2020), pages 944-953, 2020. URL: http://arxiv.org/abs/2004.04250.
  25. Shunhua Jiang, Zhao Song, Omri Weinstein, and Hengjie Zhang. Faster dynamic matrix inverse for faster LPs, 2020. URL: http://arxiv.org/abs/2004.07470.
  26. Ravindran Kannan and Santosh Vempala. Randomized algorithms in numerical linear algebra. Acta Numerica, 26:95-135, 2017. Google Scholar
  27. Iordanis Kerenidis and Anupam Prakash. Quantum recommendation systems. In Proceedings of the 8th Innovations in Theoretical Computer Science Conference (ITCS 2017), pages 49:1-49:21, 2017. URL: http://arxiv.org/abs/1603.08675.
  28. Iordanis Kerenidis and Anupam Prakash. A quantum interior point method for LPs and SDPs, 2018. URL: http://arxiv.org/abs/1808.09266.
  29. Leonid G. Khachiyan. Polynomial algorithms in linear programming. USSR Computational Mathematics and Mathematical Physics, 20(1):51-68, 1980. Google Scholar
  30. James R. Lee, Prasad Raghavendra, and David Steurer. Lower bounds on the size of semidefinite programming relaxations. In Proceedings of the 47th Annual ACM Symposium on Theory of Computing (STOC 2015). ACM, 2015. URL: http://arxiv.org/abs/1411.6317.
  31. Yin Tat Lee, Aaron Sidford, and Sam Chiu-wai Wong. A faster cutting plane method and its implications for combinatorial and convex optimization. In Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2015), pages 1049-1065. IEEE, 2015. URL: http://arxiv.org/abs/1508.04874.
  32. Michael Luby and Noam Nisan. A parallel approximation algorithm for positive linear programming. In Proceedings of the 25th Annual ACM Symposium on Theory of Computing (STOC 1993), pages 448-457. ACM, 1993. Google Scholar
  33. John E. Mitchell. Polynomial interior point cutting plane methods. Optimization Methods and Software, 18(5):507-534, 2003. Google Scholar
  34. Yurii Nesterov and Arkadi Nemirovsky. Conic formulation of a convex programming problem and duality. Optimization Methods and Software, 1(2):95-115, 1992. Google Scholar
  35. John Preskill. Quantum computing in the NISQ era and beyond. Quantum, 2:79, 2018. URL: http://arxiv.org/abs/1801.00862.
  36. Ewin Tang. Quantum-inspired classical algorithms for principal component analysis and supervised clustering, 2018. URL: http://arxiv.org/abs/1811.00414.
  37. Ewin Tang. A quantum-inspired classical algorithm for recommendation systems. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC 2019), pages 217-228. ACM, 2019. URL: http://arxiv.org/abs/1807.04271.
  38. Lieven Vandenberghe and Stephen Boyd. Semidefinite programming. SIAM Review, 38(1):49-95, 1996. Google Scholar
  39. Xiaodi Wu. Parallelized solution to semidefinite programmings in quantum complexity theory, 2010. URL: http://arxiv.org/abs/1009.2211.
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