Quantum-inspired algorithms in practice

Juan Miguel Arrazola; Alain Delgado; Bhaskar Roy Bardhan; Seth Lloyd

doi:10.22331/q-2020-08-13-307

Quantum-inspired algorithms in practice

Juan Miguel Arrazola¹, Alain Delgado¹, Bhaskar Roy Bardhan¹, and Seth Lloyd^1,2

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

Published:	2020-08-13, volume 4, page 307
Eprint:	arXiv:1905.10415v3
Doi:	https://doi.org/10.22331/q-2020-08-13-307
Citation:	Quantum 4, 307 (2020).

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

Abstract

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

@article{Arrazola2020quantuminspired,
  doi = {10.22331/q-2020-08-13-307},
  url = {https://doi.org/10.22331/q-2020-08-13-307},
  title = {Quantum-inspired algorithms in practice},
  author = {Arrazola, Juan Miguel and Delgado, Alain and Bardhan, Bhaskar Roy and Lloyd, Seth},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {4},
  pages = {307},
  month = aug,
  year = {2020}
}

► 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.
arXiv: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.
https://doi.org/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.
https://doi.org/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.
arXiv: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.
arXiv: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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
arXiv: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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
arXiv:1603.08675

[17] Seth Lloyd, Masoud Mohseni, and Patrick Rebentrost. Quantum principal component analysis. Nature Physics, 10 (9): 631, 2014. 10.1038/nphys3029.
https://doi.org/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.
https://doi.org/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.
https://doi.org/10.1109/ICEC.1996.542334

[21] Cam Nugent. S and P 500 stock data. 2018. URL https://www.kaggle.com/camnugent/sandp500.
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.
arXiv: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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
arXiv: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.
https://doi.org/10.1145/3313276.3316310

[32] Leslie G Valiant. Holographic algorithms. SIAM Journal on Computing, 37 (5): 1565–1594, 2008. 10.1137/070682575.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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] Liliya Demidova and Vladimir Maslennikov, "Thermonuclear Fusion Based Quantum-Inspired Algorithm for Solving Multiobjective Optimization Problems", Algorithms 18 12, 793 (2025).

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

[5] Barbara Collignon, "A graph-based quantum-inspired approximation of the Feynman path integral", Academia Quantum 3 2(2026).

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

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

[8] 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).

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

[10] Akshat Gaurav, Achit Katiyar, Moon Jusung, and Brij B. Gupta, Advances in Computational Intelligence and Robotics 343 (2025) ISBN:9798369380345.

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

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

[13] 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).

[14] Mohamed Ramzi Bendar, Youcef Malek, Mohamed Tadjine, and Nadjet Zioui, 2026 12th International Conference on Control, Decision and Information Technologies (CoDIT) 2103 (2026) ISBN:979-8-3195-2077-7.

[15] 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).

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

[17] 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).

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

[19] Ryan Peters, Kahn Rhrissorrakrai, Prerana Bangalore Parthasarathy, Vadim Ratner, Tanvi P. Gujarati, Meltem Tolunay, Jie Shi, Jeffrey K. Weber, Timothy A. Chan, Laxmi Parida, Sara Capponi, Filippo Utro, and Tyler J. Alban, "Quantum convolutional HLA immunogenic peptide prediction (Q-CHIPP): Next-generation neoantigen prediction with quantum neural network", Science Advances 12 30, eaec3824 (2026).

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

[21] 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).

[22] 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).

[23] Haidar Hosamo, Vagelis Plevris, Dimitrios Kraniotis, and Christian Nordahl Rolfsen, "Can quantum computing surpass classical algorithms in optimizing building performance? A benchmark study with 15,000 simulations", Energy and Buildings 346, 116156 (2025).

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

[25] Davi Geiger and Michael Werman, "Quantum Elastica", Entropy 27 4, 388 (2025).

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

[27] Torsten Pook, Jeremie Vandenplas, Juan Carlos Boschero, Esteban Aguilera, Koen Leijnse, Aneesh Chauhan, Yamine Bouzembrak, Rob Knapen, and Michael Aldridge, "Assessing the potential of quantum computing in agriculture", Computers and Electronics in Agriculture 235, 110332 (2025).

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

[29] Julian Berberich and Daniel Fink, "Quantum Computing Through the Lens of Control: A Tutorial Introduction", IEEE Control Systems 44 6, 24 (2024).

[30] 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).

[31] T Manju, P NaveenKumar, S Vinothan, and A Prashant, "A Quantum-Inspired Adaptive AI Tutor for Personalized Learning: A Quiz-Driven Knowledge Framework", EPJ Web of Conferences 360, 01020 (2026).

[32] Yunting Li, Xiaopeng Cui, Zhaoping Xiong, Bowen Liu, Bi-Ying Wang, Runqiu Shu, Nan Qiao, and Man-Hong Yung, "Quantum Molecular Docking with a Quantum-Inspired Algorithm", Journal of Chemical Theory and Computation 20 15, 6687 (2024).

[33] Youle Wang, "Near-optimal quantum kernel principal component analysis", Quantum Science and Technology 10 1, 015034 (2025).

[34] 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).

[35] Marcel Niedermeier, Adrien Moulinas, Thibaud Louvet, Jose L. Lado, and Xavier Waintal, "Solving the Gross-Pitaevskii equation on multiple different scales using the quantics tensor train representation", Physical Review Research 8 2, 023006 (2026).

[36] Liliya Demidova and Vladimir Maslennikov, "New approach to implementation of quantum-inspired genetic algorithm for numerical optimization based on high-dimensional quantum states", American Institute of Physics Conference Series PROCEEDINGS OF THE IV INTERNATIONAL CONFERENCE ON ADVANCES IN MATERIALS, EARTH SCIENCE, AND TECHNOLOGY: CAMSTECH-IV 2024 3390 1, 060028 (2026).

[37] 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.

[38] 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).

[39] Hemaswathi S, Jeya A Celin J, Sivagurunathan R, Sanjai K, Sheik Abdul Rahuman M, and Sanjaykrishnaa T, 2026 4th International Conference on Artificial Intelligence and Machine Learning Applications Theme: Healthcare and Internet of Things (AIMLA) 1 (2026) ISBN:979-8-3195-0634-4.

[40] Emelia Logah, "Efficient Machine Learning for Wi-Fi CSI-based Human Activity Recognition Using Fast Monte Carlo based Feature Extraction", EAI Endorsed Transactions on Tourism, Technology and Intelligence 2 2(2025).

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

[42] Omkar Bihani and Janez Žerovnik, "A Heuristic for Graph Coloring Based on the Ising Model", Mathematics 13 18, 2976 (2025).

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

[44] 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).

[45] Manuel H. Muñoz-Arias, Stefanos Kourtis, and Alexandre Blais, "Low-depth Clifford circuits approximately solve MaxCut", Physical Review Research 6 2, 023294 (2024).

[46] Ushaa Eswaran and Vishal Eswaran, Advances in Mechatronics and Mechanical Engineering 57 (2024) ISBN:9798369370766.

[47] Wei Zhao, Tangjie Weng, Yue Ruan, Zhi Liu, Xuangou Wu, Xiao Zheng, and Nei Kato, "Quantum Computing in Wireless Communications and Networking: A Tutorial-cum-Survey", IEEE Communications Surveys & Tutorials 27 4, 2378 (2025).

[48] Emmanuel Zambrini Cruzeiro, Christine De Mol, Serge Massar, and Stefano Pironio, "Quantum-inspired classification based on quantum state discrimination", Quantum Machine Intelligence 6 2, 79 (2024).

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

[50] 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).

[51] T. Subaranjani, P. Umarani, and V. Padmavathi, 2025 12th International Conference on Computing for Sustainable Global Development (INDIACom) 01 (2025) ISBN:978-93-80544-60-1.

[52] Naushad Ahmad and Shubham Jas, "Quantum-inspired neural networks for time-series air pollution prediction and control of the most polluted region in the world", Quantum Machine Intelligence 7 1, 9 (2025).

[53] Jinglei Cheng, Ruilin Zhou, Yuhang Gan, Chen Qian, and Junyu Liu, 2025 62nd ACM/IEEE Design Automation Conference (DAC) 1 (2025) ISBN:979-8-3315-0304-8.

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

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

[56] Pooja and Sandeep Kumar Sood, "Scientometric Analysis of Quantum Algorithms for VANET Optimization", IEEE Transactions on Systems, Man, and Cybernetics: Systems 54 10, 6447 (2024).

[57] Muhammad AbuGhanem, "Toward scalable fault-tolerant photonic quantum computers", The Journal of Supercomputing 82 2, 51 (2026).

[58] 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).

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

[60] Yutong Jiang, Zhe Zheng, Shuai S. A. Yuan, Hao Xue, Long Li, and Wei E. I. Sha, "Quantum-Inspired Optimization for 2-bit Quantized Phase-Only Metasurfaces With Near-Field Multiobjective Control", IEEE Transactions on Microwave Theory and Techniques 73 12, 9886 (2025).

[61] Rishi Goel, Casey R Myers, and Sally Shrapnel, "Quantum algorithms for causal estimands", Quantum Science and Technology 11 1, 015024 (2026).

[62] Shubh Kapoor and Vikas Garg, Quantum‐Inspired Approaches for Intelligent Data Processing 173 (2026) ISBN:9781394336418.

[63] 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).

[64] 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).

[65] 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).

[66] 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 8 2, 2300334 (2025).

[67] Markkandan S, Sharon Deena Reji, Sairam B, Shivani Narayanan, and Mridhula S, 2024 IEEE 21st India Council International Conference (INDICON) 1 (2024) ISBN:979-8-3503-9128-2.

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

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

[70] 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", Journal of Molecular Modeling 30 7, 228 (2024).

[71] Agu, Chidera Onyeka, "Quantum Computing for Explainable AI: Developing Quantum-inspired Interpretable Models for Complex Decision-making Systems", British Journal of Computer Networking and Information Technology 8 3, 1 (2025).

[72] Sanjay Kumar, Hitesh Binjrawat, and Gorav Singal, 2026 9th International Conference on Computational Intelligence in Data Science (ICCIDS) 1 (2026) ISBN:979-8-3315-6432-2.

[73] Johannes Christmann, Petr Ivashkov, Mattia Chiurco, and Guglielmo Mazzola, "From quantum-enhanced to quantum-inspired Monte Carlo", Physical Review A 111 4, 042615 (2025).

[74] Aritra Bose, Kahn Rhrissorrakrai, Filippo Utro, Laxmi Parida, Saugata Basu, Jannis Born, Sara Capponi, Dimitra Chalkia, Timothy A. Chan, Hakan Doga, Frederik F. Flöther, Gad Getz, Mark Goldsmith, Tanvi Gujarati, Aldo Guzman-Saenz, Dimitrios Iliopoulos, Gavin O. Jones, Stefan Knecht, Dhiraj Madan, Sabrina Maniscalco, Nicola Mariella, Joseph A. Morrone, Khadijeh Najafi, Pushpak Pati, Daniel Platt, Maria Anna Rapsomaniki, Anupama Ray, Omar Shehab, Ivano Tavernelli, Meltem Tolunay, Stefan Woerner, Sergiy Zhuk, and Jeannette M. Garcia, "Advancing single-cell omics and cell-based therapeutics with quantum computing", Nature Reviews Molecular Cell Biology 27 5, 394 (2026).

[75] Clement Afagwu, Hayden Gemeinhardt, Jyotsna Sharma, and Yu-Bo Sheng, "Quantum‐Enhanced Distributed Fiber‐Optic Sensing: Current State and Future Perspectives", Quantum Engineering 2026 1, 3022974 (2026).

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

[77] 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).

[78] 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).

[79] Michael Nguyen Phuc, "When Doing Nothing Is the Optimal Cyber Defense: Quantum-Inspired Abstention as a First-Class Security Action", EPJ Web of Conferences 360, 01001 (2026).

[80] Lakshya Jhajaria and Vivek Sharma, Lecture Notes in Networks and Systems 1786, 615 (2026) ISBN:978-3-032-14756-1.

[81] 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).

[82] Ishu Chaudhary, Ankesh Kumar, and KrashnKant Gupta, Quantum‐Inspired Approaches for Intelligent Data Processing 259 (2026) ISBN:9781394336418.

[83] Qian Zuo, Yimin Wei, and Hua Xiang, "Quantum-inspired algorithm for truncated total least squares solution", Journal of Computational and Applied Mathematics 451, 116042 (2024).

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

[85] 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).

[86] 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).

[87] Rita Singh and Angel Deborah Suseelan, "Quantum-Accelerated Artificial Intelligence for Edge Devices: A Review of Encodings, Models, Hybrid Architectures, and NISQ-Era Realities", Electronics 15 13, 2832 (2026).

[88] 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.

[89] 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).

[90] 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).

[91] Vladimir V. Maslennikov and Liliya A. Demidova, "Modification of a Quantum-inspired Genetic Algorithm for Numerical Optimization Using Qudit under Conditions of Simulating Quantum Decoherence", Computational nanotechnology 11 2, 58 (2024).

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

[93] Dr. Nikhil Aravind Rao, "Adaptive Quantum-Inspired Optimization Models for Large-Scale Machine Learning Workloads in Cloud Platforms", Journal of Multidisciplinary Knowledge 5 2, 233 (2025).

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

[95] Aman Kashyap, Naman Jain, and Deepak Nagar, Lecture Notes in Mechanical Engineering 457 (2026) ISBN:978-981-95-1722-0.

[96] 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).

[97] 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).

[98] 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).

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

[100] 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).

[101] Jorge de Blas, Monica Dunford, Emanuele Bagnaschi, Ayres Freitas, Pier Paolo Giardino, Christian Grefe, Michele Selvaggi, Angela Taliercio, Falk Bartels, Andrea Dainese, Cristinel Diaconu, Chiara Signorile-Signorile, Néstor Armesto, Roberta Arnaldi, Andy Buckley, David d'Enterria, Antoine Gérardin, Valentina Mantovani Sarti, Sven-Olaf Moch, Marco Pappagallo, Raimond Snellings, Urs Achim Wiedemann, Gino Isidori, Marie-Hélène Schune, Maria Laura Piscopo, Marta Calvi, Yuval Grossman, Thibaud Humair, Andreas Jüttner, Jernej F. Kamenik, Matthew Kenzie, Patrick Koppenburg, Radoslav Marchevski, Angela Papa, Guillaume Pignol, Justine Serrano, Pilar Hernandez, Sara Bolognesi, Ivan Esteban, Stephen Dolan, Valerie Domcke, Joseph Formaggio, M. C. Gonzalez-Garcia, Aart Heijboer, Aldo Ianni, Joachim Kopp, Elisa Resconi, Mark Scott, Viola Sordini, Fabio Maltoni, Rebeca Gonzalez Suarez, Benedikt Maier, Timothy Cohen, Annapaola de Cosa, Nathaniel Craig, Roberto Franceschini, Loukas Gouskos, Aurelio Juste, Sophie Renner, Lesya Shchutska, Jocelyn Monroe, Matthew McCullough, Yohei Ema, Paolo Agnes, Francesca Calore, Emanuele Castorina, Aaron Chou, Monica D'Onofrio, Maksym Ovchynnikov, Tina Pollman, Josef Pradler, Yotam Soreq, Julia Katharina Vogel, Gianluigi Arduini, Philip Burrows, Jacqueline Keintzel, Deepa Angal-Kalinin, Bernhard Auchmann, Massimo Ferrario, Angeles Faus Golfe, Roberto Losito, Anke-Susanne Mueller, Tor Raubenheimer, Marlene Turner, Pierre Vedrine, Hans Weise, Walter Wuensch, Chenghui Yu, Thomas Bergauer, Ulrich Husemann, Dorothea vom Bruch, Thea Aarrestad, Daniela Bortoletto, Shikma Bressler, Marcel Demarteau, Michael Doser, Gabriella Gaudio, Inés Gil-Botella, Andrea Giuliani, Fabrizio Palla, Rok Pestotnik, Felix Sefkow, Frank Simon, Maksym Titov, Tommaso Boccali, Borut Kersevan, Daniel Murnane, Gonzalo Merino Arevalo, John Derek Chapman, Frank-Dieter Gaede, Stefano Giagu, Maria Girone, Heather M. Gray, Giovanni Iadarola, Stephane Jezequel, Gregor Kasieczka, David Lange, Sinéad M. Ryan, Nicole Skidmore, Sofia Vallecorsa, Eric Laenen, Anadi Canepa, Xinchou Lou, Rogerio Rosenfeld, Yuji Yamazaki, Roger Forty, Karl Jakobs, Hugh Montgomery, Mike Seidel, and Paris Sphicas, "Physics Briefing Book: Input for the 2026 update of the European Strategy for Particle Physics", arXiv:2511.03883, (2025).

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

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

[104] Tariq M. Khan and Antonio Robles-Kelly, "Machine Learning: Quantum vs Classical", IEEE Access 8, 219275 (2020).

[105] 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).

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

[107] Chen Ding, Tian-Yi Bao, and He-Liang Huang, "Quantum-Inspired Support Vector Machine", arXiv:1906.08902, (2019).

[108] 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).

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

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

[111] 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).

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

[113] 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).

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

[115] Ryan LaRose, "A brief history of quantum vs classical computational advantage", arXiv:2412.14703, (2024).

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

[117] 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).

[118] 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).

[119] S. Golestan, M. R. Habibi, S. Y. Mousazadeh Mousavi, J. M. Guerrero, and J. C. Vasquez, "Quantum computation in power systems: An overview of recent advances", Energy Reports 9, 584 (2023).

[120] Jirawat Tangpanitanon, "Quantum Computing for Energy Management: A Semi Non-Technical Guide for Practitioners", arXiv:2411.11901, (2024).

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

[122] 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).

[123] 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).

[124] 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).

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

[126] 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).

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

[128] Shaozhi Li, M Sabbir Salek, Yao Wang, and Mashrur Chowdhury, "Quantum-inspired activation functions and quantum Chebyshev-polynomial network", arXiv:2404.05901, (2024).

[129] 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).

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

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

[132] Aritra Das, Joseph T. Iosue, and Victor V. Albert, "Quantum-inspired Benchmark for Estimating Intrinsic Dimension", arXiv:2510.01335, (2025).

[133] 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).

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

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

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

[137] Srikar Kasi, John Kaewell, and Kyle Jamieson, "A Quantum Annealer-Enabled Decoder and Hardware Topology for NextG Wireless Polar Codes", IEEE Transactions on Wireless Communications 23 4, 3780 (2024).

[138] Matthias Tschöpe, Vitor Fortes Rey, Sogo Pierre Sanon, Paul Lukowicz, Nikolaos Palaiodimopoulos, and Maximilian Kiefer-Emmanouilidis, "Boosting Classification with Quantum-Inspired Augmentations", arXiv:2506.22241, (2025).

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

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

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

[142] SeungYeop Baik, Sicheol Sung, and Yo-Sub Han, "A Framework for Quantum Finite-State Languages with Density Mapping", arXiv:2407.02776, (2024).

[143] 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).

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

[145] 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).

[146] Shlomi Debi and Adi Makmal, "Variational quantum recommendation system with embedded latent vectors", Scientific Reports 15 1, 32907 (2025).

[147] Ryo Sakai and Chen-Yu Liu, "Tensor Network Generator-Enhanced Optimization for Traveling Salesman Problem", arXiv:2602.20175, (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-10 03:35:52) and SAO/NASA ADS (last updated successfully 2026-08-09 15:31:22). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch ADS cited-by data during last attempt 2026-08-10 03:35:52: Cannot retrieve data from ADS due to rate limitations.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.