Quantum-inspired algorithms in practice

Juan Miguel Arrazola1, Alain Delgado1, Bhaskar Roy Bardhan1, and Seth Lloyd1,2

1Xanadu, Toronto, Ontario, M5G 2C8, Canada
2Massachusetts Institute of Technology, Department of Mechanical Engineering, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


We study the practical performance of quantum-inspired algorithms for recommendation systems and linear systems of equations. These algorithms were shown to have an exponential asymptotic speedup compared to previously known classical methods for problems involving low-rank matrices, but with complexity bounds that exhibit a hefty polynomial overhead compared to quantum algorithms. This raised the question of whether these methods were actually useful in practice. We conduct a theoretical analysis aimed at identifying their computational bottlenecks, then implement and benchmark the algorithms on a variety of problems, including applications to portfolio optimization and movie recommendations. On the one hand, our analysis reveals that the performance of these algorithms is better than the theoretical complexity bounds would suggest. On the other hand, their performance as seen in our implementation degrades noticeably as the rank and condition number of the input matrix are increased. Overall, our results indicate that quantum-inspired algorithms can perform well in practice provided that stringent conditions are met: low rank, low condition number, and very large dimension of the input matrix. By contrast, practical datasets are often sparse and high-rank, precisely the type that can be handled by quantum algorithms.

Please see this blog post for a summary of the work.

► BibTeX data

► References

[1] Boaz Barak, Ankur Moitra, Ryan O'Donnell, Prasad Raghavendra, Oded Regev, David Steurer, Luca Trevisan, Aravindan Vijayaraghavan, David Witmer, and John Wright. Beating the random assignment on constraint satisfaction problems of bounded degree. arXiv:1505.03424, 2015. URL https:/​/​arxiv.org/​abs/​1505.03424.

[2] Jacob Biamonte, Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe, and Seth Lloyd. Quantum machine learning. Nature, 549 (7671): 195, 2017. 10.1038/​nature23474.

[3] L Chakhmakhchyan, NJ Cerf, and R Garcia-Patron. Quantum-inspired algorithm for estimating the permanent of positive semidefinite matrices. Physical Review A, 96 (2): 022329, 2017. 10.1103/​PhysRevA.96.022329.

[4] Nai-Hui Chia, Han-Hsuan Lin, and Chunhao Wang. Quantum-inspired sublinear classical algorithms for solving low-rank linear systems. arXiv:1811.04852, 2018. URL https:/​/​arxiv.org/​abs/​1811.04852.

[5] Nai-Hui Chia, Tongyang Li, Han-Hsuan Lin, and Chunhao Wang. Quantum-inspired classical sublinear-time algorithm for solving low-rank semidefinite programming via sampling approaches. arXiv:1901.03254, 2019. URL https:/​/​arxiv.org/​abs/​1901.03254.

[6] AV Abs da Cruz, Marley Maria Bernardes Rebuzzi Vellasco, and Marco Aurélio Cavalcanti Pacheco. Quantum-inspired evolutionary algorithm for numerical optimization. In Hybrid evolutionary algorithms, pages 19–37. Springer, 2007. 10.1007/​978-3-540-73297-6_2.

[7] Yogesh Dahiya, Dimitris Konomis, and David P Woodruff. An empirical evaluation of sketching for numerical linear algebra. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pages 1292–1300. ACM, 2018. 10.1145/​3219819.3220098.

[8] Petros Drineas, Ravi Kannan, and Michael W Mahoney. Fast monte carlo algorithms for matrices ii: Computing a low-rank approximation to a matrix. SIAM Journal on computing, 36 (1): 158–183, 2006. 10.1137/​S0097539704442696.

[9] Alan Frieze, Ravi Kannan, and Santosh Vempala. Fast monte-carlo algorithms for finding low-rank approximations. Journal of the ACM (JACM), 51 (6): 1025–1041, 2004. 10.1145/​1039488.1039494.

[10] András Gilyén, Seth Lloyd, and Ewin Tang. Quantum-inspired low-rank stochastic regression with logarithmic dependence on the dimension. arXiv:1811.04909, 2018. URL https:/​/​arxiv.org/​abs/​1811.04909.

[11] Nathan Halko, Per-Gunnar Martinsson, and Joel A Tropp. Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. SIAM review, 53 (2): 217–288, 2011. 10.1137/​090771806.

[12] Kuk-Hyun Han and Jong-Hwan Kim. Quantum-inspired evolutionary algorithm for a class of combinatorial optimization. IEEE transactions on evolutionary computation, 6 (6): 580–593, 2002. 10.1109/​TEVC.2002.804320.

[13] F Maxwell Harper and Joseph A Konstan. The movielens datasets: History and context. ACM Transactions on Interactive Intelligent Systems (TIIS), 5 (4): 19, 2016. 10.1145/​2827872.

[14] Aram W Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum algorithm for linear systems of equations. Physical Review Letters, 103 (15): 150502, 2009. 10.1103/​PhysRevLett.103.150502.

[15] Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. Journal of Computer and System Sciences, 62 (2): 367–375, 2001. 10.1006/​jcss.2000.1727.

[16] Iordanis Kerenidis and Anupam Prakash. Quantum recommendation systems. arXiv:1603.08675, 2016. URL https:/​/​arxiv.org/​abs/​1603.08675.

[17] Seth Lloyd, Masoud Mohseni, and Patrick Rebentrost. Quantum principal component analysis. Nature Physics, 10 (9): 631, 2014. 10.1038/​nphys3029.

[18] Harry Markowitz. Portfolio selection. The Journal of Finance, 7 (1): 77–91, 1952.

[19] Per-Gunnar Martinsson, Vladimir Rokhlin, and Mark Tygert. A randomized algorithm for the decomposition of matrices. Applied and Computational Harmonic Analysis, 30 (1): 47–68, 2011. 10.1016/​j.acha.2007.12.002.

[20] Ajit Narayanan and Mark Moore. Quantum-inspired genetic algorithms. In Evolutionary Computation, 1996., Proceedings of IEEE International Conference on, pages 61–66. IEEE, 1996. 10.1109/​ICEC.1996.542334.

[21] Cam Nugent. S and P 500 stock data. 2018. URL https:/​/​www.kaggle.com/​camnugent/​sandp500.

[22] Patrick Rebentrost and Seth Lloyd. Quantum computational finance: quantum algorithm for portfolio optimization. arXiv:1811.03975, 2018. URL https:/​/​arxiv.org/​abs/​1811.03975.

[23] Patrick Rebentrost, Masoud Mohseni, and Seth Lloyd. Quantum support vector machine for big data classification. Physical Review Letters, 113 (13): 130503, 2014. 10.1103/​PhysRevLett.113.130503.

[24] Vladimir Rokhlin and Mark Tygert. A fast randomized algorithm for overdetermined linear least-squares regression. Proceedings of the National Academy of Sciences, 105 (36): 13212–13217, 2008. 10.1073/​pnas.0804869105.

[25] Vladimir Rokhlin, Arthur Szlam, and Mark Tygert. A randomized algorithm for principal component analysis. SIAM Journal on Matrix Analysis and Applications, 31 (3): 1100–1124, 2009. 10.1137/​080736417.

[26] Troels F Rønnow, Zhihui Wang, Joshua Job, Sergio Boixo, Sergei V Isakov, David Wecker, John M Martinis, Daniel A Lidar, and Matthias Troyer. Defining and detecting quantum speedup. Science, 345 (6195): 420–424, 2014. 10.1126/​science.1252319.

[27] Tamas Sarlos. Improved approximation algorithms for large matrices via random projections. In Foundations of Computer Science, 2006. FOCS'06. 47th Annual IEEE Symposium on, pages 143–152. IEEE, 2006. 10.1109/​FOCS.2006.37.

[28] Jianhong Shen. On the singular values of Gaussian random matrices. Linear Algebra and its Applications, 326 (1): 1 – 14, 2001. ISSN 0024-3795. 10.1016/​S0024-3795(00)00322-0.

[29] Allan Sly and Nike Sun. The computational hardness of counting in two-spin models on d-regular graphs. In Foundations of Computer Science (FOCS), 2012 IEEE 53rd Annual Symposium on, pages 361–369. IEEE, 2012. 10.1109/​FOCS.2012.56.

[30] Ewin Tang. Quantum-inspired classical algorithms for principal component analysis and supervised clustering. arXiv:1811.00414, 2018. URL https:/​/​arxiv.org/​abs/​1811.00414.

[31] Ewin Tang. A quantum-inspired classical algorithm for recommendation systems. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 217–228, 2019. 10.1145/​3313276.3316310.

[32] Leslie G Valiant. Holographic algorithms. SIAM Journal on Computing, 37 (5): 1565–1594, 2008. 10.1137/​070682575.

[33] David P Woodruff et al. Sketching as a tool for numerical linear algebra. Foundations and Trends® in Theoretical Computer Science, 10 (1–2): 1–157, 2014. 10.1561/​0400000060.

[34] Franco Woolfe, Edo Liberty, Vladimir Rokhlin, and Mark Tygert. A fast randomized algorithm for the approximation of matrices. Applied and Computational Harmonic Analysis, 25 (3): 335–366, 2008. 10.1016/​j.acha.2007.12.002.

Cited by

[1] Srikar Kasi and Kyle Jamieson, Proceedings of the 26th Annual International Conference on Mobile Computing and Networking 1 (2020) ISBN:9781450370851.

[2] Nai-Hui Chia, Tongyang Li, Han-Hsuan Lin, and Chunhao Wang, "Quantum-inspired sublinear algorithm for solving low-rank semidefinite programming", arXiv:1901.03254.

[3] Jonathan Allcock, Chang-Yu Hsieh, Iordanis Kerenidis, and Shengyu Zhang, "Quantum algorithms for feedforward neural networks", arXiv:1812.03089.

[4] P. A. M. Casares and M. A. Martin-Delgado, "A Quantum Interior-Point Predictor-Corrector Algorithm for Linear Programming", arXiv:1902.06749.

[5] 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", arXiv:1910.06151.

[6] Quntao Zhuang and Zheshen Zhang, "Physical-Layer Supervised Learning Assisted by an Entangled Sensor Network", Physical Review X 9 4, 041023 (2019).

[7] M. Cerezo, Kunal Sharma, Andrew Arrasmith, and Patrick J. Coles, "Variational Quantum State Eigensolver", arXiv:2004.01372.

[8] John Realpe-Gómez and Nathan Killoran, "Quantum-inspired memory-enhanced stochastic algorithms", arXiv:1906.00263.

[9] Lin Lin and Yu Tong, "Optimal quantum eigenstate filtering with application to solving quantum linear systems", arXiv:1910.14596.

[10] Dhawal Jethwani, François Le Gall, and Sanjay K. Singh, "Quantum-Inspired Classical Algorithms for Singular Value Transformation", arXiv:1910.05699.

[11] Chen Ding, Tian-Yi Bao, and He-Liang Huang, "Quantum-Inspired Support Vector Machine", arXiv:1906.08902.

[12] Iordanis Kerenidis, Jonas Landman, and Anupam Prakash, "Quantum Algorithms for Deep Convolutional Neural Networks", arXiv:1911.01117.

[13] Iordanis Kerenidis, Alessandro Luongo, and Anupam Prakash, "Quantum Expectation-Maximization for Gaussian Mixture Models", arXiv:1908.06657.

[14] Lucas Lamata, "Quantum machine learning and quantum biomimetics: A perspective", arXiv:2004.12076.

[15] Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, and Dacheng Tao, "Quantum-inspired algorithm for general minimum conical hull problems", Physical Review Research 2 3, 033199 (2020).

[16] Hayata Yamasaki, Kosuke Fukui, Yuki Takeuchi, Seiichiro Tani, and Masato Koashi, "Polylog-overhead highly fault-tolerant measurement-based quantum computation: all-Gaussian implementation with Gottesman-Kitaev-Preskill code", arXiv:2006.05416.

[17] X. -L. Ouyang, X. -Z. Huang, Y. -K. Wu, W. -G. Zhang, X. Wang, H. -L. Zhang, L. He, X. -Y. Chang, and L. -M. Duan, "Experimental demonstration of quantum-enhanced machine learning in a nitrogen-vacancy-center system", Physical Review A 101 1, 012307 (2020).

[18] Wen Guan, Gabriel Perdue, Arthur Pesah, Maria Schuld, Koji Terashi, Sofia Vallecorsa, and Jean-Roch Vlimant, "Quantum Machine Learning in High Energy Physics", arXiv:2005.08582.

[19] Yu Tong, Dong An, Nathan Wiebe, and Lin Lin, "Fast inversion, preconditioned quantum linear system solvers, and fast evaluation of matrix functions", arXiv:2008.13295.

[20] Philip Easom-McCaldin, Ahmed Bouridane, Ammar Belatreche, and Richard Jiang, "Towards Building A Facial Identification System Using Quantum Machine Learning Techniques", arXiv:2008.12616.

[21] Jonathan Allcock and Chang-Yu Hsieh, "Quantum algorithm for training nonlinear SVMs in almost linear time", arXiv:2006.10299.

[22] Bojia Duan, Jiabin Yuan, Chao-Hua Yu, Jianbang Huang, and Chang-Yu Hsieh, "A survey on HHL algorithm: From theory to application in quantum machine learning", Physics Letters A 384, 126595 (2020).

[23] Iordanis Kerenidis and Alessandro Luongo, "Classification of the MNIST data set with quantum slow feature analysis", Physical Review A 101 6, 062327 (2020).

[24] Almudena Carrera Vazquez, Ralf Hiptmair, and Stefan Woerner, "Enhancing the Quantum Linear Systems Algorithm using Richardson Extrapolation", arXiv:2009.04484.

[25] Viraj Kulkarni, Milind Kulkarni, and Aniruddha Pant, "Quantum Computing Methods for Supervised Learning", arXiv:2006.12025.

[26] A. S. Boev, A. S. Rakitko, S. R. Usmanov, A. N. Kobzeva, I. V. Popov, V. V. Ilinsky, E. O. Kiktenko, and A. K. Fedorov, "Genome assembly using quantum and quantum-inspired annealing", arXiv:2004.06719.

[27] Srikar Kasi and Kyle Jamieson, "Towards Quantum Belief Propagation for LDPC Decoding in Wireless Networks", arXiv:2007.11069.

The above citations are from Crossref's cited-by service (last updated successfully 2020-09-22 11:12:25) and SAO/NASA ADS (last updated successfully 2020-09-22 11:12:26). The list may be incomplete as not all publishers provide suitable and complete citation data.