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] Nic Ezzell, Elliott M Ball, Aliza U Siddiqui, Mark M Wilde, Andrew T Sornborger, Patrick J Coles, and Zoë Holmes, "Quantum mixed state compiling", Quantum Science and Technology 8 3, 035001 (2023).

[2] Youle Wang and Yu Luo, "Resource-efficient quantum principal component analysis", Quantum Science and Technology 9 3, 035031 (2024).

[3] Seyran Saeedi, Aliakbar Panahi, and Tom Arodz, "Quantum semi-supervised kernel learning", Quantum Machine Intelligence 3 2, 24 (2021).

[4] Arvindhan Muthusamy, Advances in Computer and Electrical Engineering 22 (2023) ISBN:9781668475355.

[5] Erik J. Gustafson, Henry Lamm, and Judah Unmuth-Yockey, "Quantum mean estimation for lattice field theory", Physical Review D 107 11, 114511 (2023).

[6] Qisheng Wang, Zhicheng Zhang, Kean Chen, Ji Guan, Wang Fang, Junyi Liu, and Mingsheng Ying, "Quantum Algorithm for Fidelity Estimation", IEEE Transactions on Information Theory 69 1, 273 (2023).

[7] Sebastián Roca-Jerat, Juan Román-Roche, and David Zueco, "Qudit machine learning", Machine Learning: Science and Technology 5 1, 015057 (2024).

[8] Kei Majima and Naoko Koide-Majima, "Quantum-Inspired Algorithms for Accelerating Machine Learning", The Brain & Neural Networks 29 4, 186 (2022).

[9] Lin Lin and Yu Tong, "Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems", Quantum 4, 361 (2020).

[10] Xiaoqiang Wang, Lejia Gu, Heung-wing Lee, and Guofeng Zhang, "Quantum context-aware recommendation systems based on tensor singular value decomposition", Quantum Information Processing 20 5, 190 (2021).

[11] Yu Tong, Dong An, Nathan Wiebe, and Lin Lin, "Fast inversion, preconditioned quantum linear system solvers, fast Green's-function computation, and fast evaluation of matrix functions", Physical Review A 104 3, 032422 (2021).

[12] Maria Schuld and Francesco Petruccione, Quantum Science and Technology 1 (2021) ISBN:978-3-030-83097-7.

[13] 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", Scientific Reports 11 1, 13183 (2021).

[14] Shihao Zhang and Lvzhou Li, "A brief introduction to quantum algorithms", CCF Transactions on High Performance Computing 4 1, 53 (2022).

[15] Barry C. Sanders, "Quantum computing for data science", Journal of Physics: Conference Series 2438 1, 012007 (2023).

[16] Kodai Shiba, Chih-Chieh Chen, Masaru Sogabe, Katsuyoshi Sakamoto, and Tomah Sogabe, "Quantum-Inspired Classification Algorithm from DBSCAN–Deutsch–Jozsa Support Vectors and Ising Prediction Model", Applied Sciences 11 23, 11386 (2021).

[17] Elena Peña Tapia, Giannicola Scarpa, and Alejandro Pozas-Kerstjens, "A didactic approach to quantum machine learning with a single qubit", Physica Scripta 98 5, 054001 (2023).

[18] Vedran Dunjko and Peter Wittek, "A non-review of Quantum Machine Learning: trends and explorations", Quantum Views 4, 32 (2020).

[19] András Gilyén, Zhao Song, and Ewin Tang, "An improved quantum-inspired algorithm for linear regression", Quantum 6, 754 (2022).

[20] Hsin-Yuan Huang, Richard Kueng, and John Preskill, "Information-Theoretic Bounds on Quantum Advantage in Machine Learning", Physical Review Letters 126 19, 190505 (2021).

[21] Chen Ding, Tian-Yi Bao, and He-Liang Huang, "Quantum-Inspired Support Vector Machine", IEEE Transactions on Neural Networks and Learning Systems 33 12, 7210 (2022).

[22] Alokananda Dey, Sandip Dey, Siddhartha Bhattacharyya, Jan Platos, and Vaclav Snasel, "Quantum inspired meta‐heuristic approaches for automatic clustering of colour images", International Journal of Intelligent Systems 36 9, 4852 (2021).

[23] Hermanni Heimonen, Adrian Auer, Ville Bergholm, Inés de Vega, and Mikko Möttönen, Computational Methods in Applied Sciences 58, 373 (2023) ISBN:978-3-031-29081-7.

[24] Alonso Viladomat Jasso, Ark Modi, Roberto Ferrara, Christian Deppe, Janis Nötzel, Fred Fung, and Maximilian Schädler, "Quantum and Quantum-Inspired Stereographic K Nearest-Neighbour Clustering", Entropy 25 9, 1361 (2023).

[25] Armando Bellante, Alessandro Luongo, and Stefano Zanero, "Quantum algorithms for SVD-based data representation and analysis", Quantum Machine Intelligence 4 2, 20 (2022).

[26] Sebastian M. Luber and Julia Binder, Chancen und Risiken von Quantentechnologien 125 (2022) ISBN:978-3-658-37533-1.

[27] Dylan Herman, Cody Googin, Xiaoyuan Liu, Yue Sun, Alexey Galda, Ilya Safro, Marco Pistoia, and Yuri Alexeev, "Quantum computing for finance", Nature Reviews Physics 5 8, 450 (2023).

[28] Abhishek Gunjan, Siddhartha Bhattacharyya, and Aboul Ella Hassanien, Smart Innovation, Systems and Technologies 358, 665 (2023) ISBN:978-981-99-3415-7.

[29] Max Hunter Gordon, M. Cerezo, Lukasz Cincio, and Patrick J. Coles, "Covariance Matrix Preparation for Quantum Principal Component Analysis", PRX Quantum 3 3, 030334 (2022).

[30] M. Cerezo, Kunal Sharma, Andrew Arrasmith, and Patrick J. Coles, "Variational quantum state eigensolver", npj Quantum Information 8 1, 113 (2022).

[31] Andrew Jackson, Theodoros Kapourniotis, and Animesh Datta, "Partition-function estimation: Quantum and quantum-inspired algorithms", Physical Review A 107 1, 012421 (2023).

[32] Roberto Giuntini, Andrés Camilo Granda Arango, Hector Freytes, Federico Hernan Holik, and Giuseppe Sergioli, "Multi-class classification based on quantum state discrimination", Fuzzy Sets and Systems 467, 108509 (2023).

[33] Jonathan Allcock and Chang-Yu Hsieh, "A quantum extension of SVM-perf for training nonlinear SVMs in almost linear time", Quantum 4, 342 (2020).

[34] Teresa Sancho-Lorente, Juan Román-Roche, and David Zueco, "Quantum kernels to learn the phases of quantum matter", Physical Review A 105 4, 042432 (2022).

[35] Hideyuki Miyahara, Yiyou Chen, Vwani Roychowdhury, and Louis-Serge Bouchard, "Decoherence mitigation by embedding a logical qubit in a qudit", Quantum Information Processing 22 7, 278 (2023).

[36] Po-Hsun Fang, Yan-Syun Chen, Jhih-Sheng Wu, and Peichen Yu, "Inverse Reticle Optimization With Quantum Annealing and Hybrid Solvers", IEEE Access 12, 33069 (2024).

[37] Helen Urgelles, Shivam Maheshwari, Swaraj Shekhar Nande, Riccardo Bassoli, Frank H.P. Fitzek, and Jose F. Monserrat, "In‐Network Quantum Computing for Future 6G Networks", Advanced Quantum Technologies 2300334 (2024).

[38] S. Mangini, F. Tacchino, D. Gerace, D. Bajoni, and C. Macchiavello, "Quantum computing models for artificial neural networks", Europhysics Letters 134 1, 10002 (2021).

[39] Giovanni Pilato and Filippo Vella, "A Survey on Quantum Computing for Recommendation Systems", Information 14 1, 20 (2022).

[40] Changpeng Shao and Ashley Montanaro, "Faster Quantum-inspired Algorithms for Solving Linear Systems", ACM Transactions on Quantum Computing 3 4, 1 (2022).

[41] Shiqin Di, Jinchen Xu, Guoqiang Shu, Congcong Feng, Xiaodong Ding, and Zheng Shan, "Amplitude transformed quantum convolutional neural network", Applied Intelligence 53 18, 20863 (2023).

[42] Pan Zou, Jianxin (Roger) Jiao, and Feng Zhou, "A twofold update quantum-inspired genetic algorithm for efficient combinatorial optimal decisions in engineering system design and operations", Journal of Engineering Design 34 4, 271 (2023).

[43] Seyed Shakib Vedaie, Archismita Dalal, Eduardo J. Páez, and Barry C. Sanders, "Framework for learning and control in the classical and quantum domains", Annals of Physics 458, 169471 (2023).

[44] Lirandë Pira and Chris Ferrie, "An invitation to distributed quantum neural networks", Quantum Machine Intelligence 5 2, 23 (2023).

[45] Francesco Slongo, Philipp Hauke, Pietro Faccioli, and Cristian Micheletti, "Quantum-inspired encoding enhances stochastic sampling of soft matter systems", Science Advances 9 43, eadi0204 (2023).

[46] Ahmed A. Abd El-Latif, Bassem Abd-El-Atty, Mohamed Amin, and Abdullah M. Iliyasu, "Quantum-inspired cascaded discrete-time quantum walks with induced chaotic dynamics and cryptographic applications", Scientific Reports 10 1, 1930 (2020).

[47] Vinooth Rao Kulkarni, Daniel Chen, Shuai Xu, Qiang Guan, and Vipin Chaudhary, Smart Innovation, Systems and Technologies 333, 29 (2023) ISBN:978-981-19-8093-0.

[48] Louis-Paul Henry, Slimane Thabet, Constantin Dalyac, and Loïc Henriet, "Quantum evolution kernel: Machine learning on graphs with programmable arrays of qubits", Physical Review A 104 3, 032416 (2021).

[49] Nai-Hui Chia, András Pal Gilyén, Tongyang Li, Han-Hsuan Lin, Ewin Tang, and Chunhao Wang, "Sampling-based Sublinear Low-rank Matrix Arithmetic Framework for Dequantizing Quantum Machine Learning", Journal of the ACM 69 5, 1 (2022).

[50] Ewin Tang, "Quantum Principal Component Analysis Only Achieves an Exponential Speedup Because of Its State Preparation Assumptions", Physical Review Letters 127 6, 060503 (2021).

[51] Viraj Kulkarni, Milind Kulkarni, and Aniruddha Pant, "Quantum computing methods for supervised learning", Quantum Machine Intelligence 3 2, 23 (2021).

[52] Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S. Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Alán Aspuru-Guzik, "Noisy intermediate-scale quantum algorithms", Reviews of Modern Physics 94 1, 015004 (2022).

[53] Alexander M. Dalzell, Sam McArdle, Mario Berta, Przemyslaw Bienias, Chi-Fang Chen, András Gilyén, Connor T. Hann, Michael J. Kastoryano, Emil T. Khabiboulline, Aleksander Kubica, Grant Salton, Samson Wang, and Fernando G. S. L. Brandão, "Quantum algorithms: A survey of applications and end-to-end complexities", arXiv:2310.03011, (2023).

[54] 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).

[55] Larry Huynh, Jin Hong, Ajmal Mian, Hajime Suzuki, Yanqiu Wu, and Seyit Camtepe, "Quantum-Inspired Machine Learning: a Survey", arXiv:2308.11269, (2023).

[56] Ewin Tang, "Quantum principal component analysis only achieves an exponential speedup because of its state preparation assumptions", arXiv:1811.00414, (2018).

[57] He-Liang Huang, Xiao-Yue Xu, Chu Guo, Guojing Tian, Shi-Jie Wei, Xiaoming Sun, Wan-Su Bao, and Gui-Lu Long, "Near-term quantum computing techniques: Variational quantum algorithms, error mitigation, circuit compilation, benchmarking and classical simulation", Science China Physics, Mechanics, and Astronomy 66 5, 250302 (2023).

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

[59] Iordanis Kerenidis, Jonas Landman, and Anupam Prakash, "Quantum Algorithms for Deep Convolutional Neural Networks", arXiv:1911.01117, (2019).

[60] 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, (2019).

[61] Dhawal Jethwani, François Le Gall, and Sanjay K. Singh, "Quantum-Inspired Classical Algorithms for Singular Value Transformation", arXiv:1910.05699, (2019).

[62] 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, (2020).

[63] András Gilyén, Zhao Song, and Ewin Tang, "An improved quantum-inspired algorithm for linear regression", arXiv:2009.07268, (2020).

[64] Dmitry A. Chermoshentsev, Aleksei O. Malyshev, Mert Esencan, Egor S. Tiunov, Douglas Mendoza, Alán Aspuru-Guzik, Aleksey K. Fedorov, and Alexander I. Lvovsky, "Polynomial unconstrained binary optimisation inspired by optical simulation", arXiv:2106.13167, (2021).

[65] 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).

[66] Changpeng Shao and Ashley Montanaro, "Faster quantum-inspired algorithms for solving linear systems", arXiv:2103.10309, (2021).

[67] Nai-Hui Chia, Tongyang Li, Han-Hsuan Lin, and Chunhao Wang, "Quantum-inspired sublinear algorithm for solving low-rank semidefinite programming", arXiv:1901.03254, (2019).

[68] Philip Easom-McCaldin, Ahmed Bouridane, Ammar Belatreche, and Richard Jiang, "Towards Building A Facial Identification System Using Quantum Machine Learning Techniques", arXiv:2008.12616, (2020).

[69] Bobak Kiani, Randall Balestriero, Yann LeCun, and Seth Lloyd, "projUNN: efficient method for training deep networks with unitary matrices", arXiv:2203.05483, (2022).

[70] 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, (2020).

[71] Almudena Carrera Vazquez, Ralf Hiptmair, and Stefan Woerner, "Enhancing the Quantum Linear Systems Algorithm using Richardson Extrapolation", arXiv:2009.04484, (2020).

[72] Ark Modi, Alonso Viladomat Jasso, Roberto Ferrara, Christian Deppe, Janis Noetzel, Fred Fung, and Maximilian Schaedler, "A Review of the Applications of Quantum Machine Learning in Optical Communication Systems", arXiv:2309.05205, (2023).

[73] Shaozhi Li, M Sabbir Salek, Yao Wang, and Mashrur Chowdhury, "Quantum-inspired activation functions in the convolutional neural network", arXiv:2404.05901, (2024).

[74] Jonathan Allcock, Chang-Yu Hsieh, Iordanis Kerenidis, and Shengyu Zhang, "Quantum algorithms for feedforward neural networks", arXiv:1812.03089, (2018).

[75] P. A. M. Casares and M. A. Martin-Delgado, "A quantum interior-point predictor-corrector algorithm for linear programming", Journal of Physics A Mathematical General 53 44, 445305 (2020).

[76] Qian Zuo and Tongyang Li, "Fast and Practical Quantum-Inspired Classical Algorithms for Solving Linear Systems", arXiv:2307.06627, (2023).

[77] Leonard Wossnig, "Quantum Machine Learning For Classical Data", arXiv:2105.03684, (2021).

[78] Jonas Landman, "Quantum Algorithms for Unsupervised Machine Learning and Neural Networks", arXiv:2111.03598, (2021).

[79] Elias Starchl and Helmut Ritsch, "Unraveling the origin of higher success probabilities in quantum annealing versus semi-classical annealing", Journal of Physics B Atomic Molecular Physics 55 2, 025501 (2022).

[80] Iordanis Kerenidis, Alessandro Luongo, and Anupam Prakash, "Quantum Expectation-Maximization for Gaussian Mixture Models", arXiv:1908.06657, (2019).

[81] 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).

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

[83] Daniel Chen, Yekun Xu, Betis Baheri, Chuan Bi, Ying Mao, Qiang Quan, and Shuai Xu, "Quantum-Inspired Classical Algorithm for Principal Component Regression", arXiv:2010.08626, (2020).

[84] Viraj Kulkarni, Milind Kulkarni, and Aniruddha Pant, "Quantum Computing Methods for Supervised Learning", arXiv:2006.12025, (2020).

[85] Srikar Kasi and Kyle Jamieson, "Towards Quantum Belief Propagation for LDPC Decoding in Wireless Networks", arXiv:2007.11069, (2020).

[86] Astrid Bötticher, Zeki C. Seskir, and Johannes Ruhland, "Introducing a Research Program for Quantum Humanities: Applications", arXiv:2303.05457, (2023).

[87] John Realpe-Gómez and Nathan Killoran, "Quantum-inspired memory-enhanced stochastic algorithms", arXiv:1906.00263, (2019).

[88] Seyran Saeedi, Aliakbar Panahi, and Tom Arodz, "Quantum Semi-Supervised Kernel Learning", arXiv:2204.10700, (2022).

[89] Lucas Lamata, "Quantum machine learning and quantum biomimetics: A perspective", arXiv:2004.12076, (2020).

[90] Yunting Li, Xiaopeng Cui, Zhaoping Xiong, Zuoheng Zou, Bowen Liu, Bi-Ying Wang, Runqiu Shu, Huangjun Zhu, Nan Qiao, and Man-Hong Yung, "Efficient molecular conformation generation with quantum-inspired algorithm", arXiv:2404.14101, (2024).

[91] Daniel Chen, Yekun Xu, Betis Baheri, Samuel A. Stein, Chuan Bi, Ying Mao, Qiang Quan, and Shuai Xu, "Quantum-Inspired Classical Algorithm for Slow Feature Analysis", arXiv:2012.00824, (2020).

[92] Naoko Koide-Majima and Kei Majima, "Quantum-inspired canonical correlation analysis for exponentially large dimensional data", arXiv:1907.03236, (2019).

The above citations are from Crossref's cited-by service (last updated successfully 2024-05-26 03:46:59) and SAO/NASA ADS (last updated successfully 2024-05-26 03:47:01). The list may be incomplete as not all publishers provide suitable and complete citation data.