Abstract
Clustering is one of the most crucial problems in unsupervised learning, and the well-known k-means algorithm can be implemented on a quantum computer with a significant speedup. However, for the clustering problems that cannot be solved using the k-means algorithm, a powerful method called spectral clustering is used. In this study, we propose a circuit design to implement spectral clustering on a quantum processor with substantial speedup by initializing the processor into a maximally entangled state and encoding the data information into an efficiently simulatable Hamiltonian. Compared to the established quantum k-means algorithms, our method does not require a quantum random access memory or a quantum adiabatic process. It relies on an appropriate embedding of quantum phase estimation into Grover’s search to gain the quantum speedup. Simulations demonstrate that our method effectively solves clustering problems and is an important supplement to quantum k-means algorithm for unsupervised learning.
Similar content being viewed by others
References
Biamonte J, Wittek P, Pancotti N, et al. Quantum machine learning. Nature, 2017, 549: 195–202
Wiebe N, Braun D, Lloyd S. Quantum algorithm for data fitting. Phys Rev Lett, 2012, 109: 050505
Rebentrost P, Mohseni M, Lloyd S. Quantum support vector machine for big data classification. Phys Rev Lett, 2014, 113: 130503
Ye Z K, Li L Z, Situ H Z, et al. Quantum speedup of twin support vector machines. Sci China Inf Sci, 2020, 63: 189501
Lloyd S, Mohseni M, Rebentrost P. Quantum principal component analysis. Nat Phys, 2014, 10: 631–633
Amin M H, Andriyash E, Rolfe J, et al. Quantum Boltzmann machine. Phys Rev X, 2018, 8: 021050
Dong D Y, Chen C L, Li H X, et al. Quantum reinforcement learning. IEEE Trans Syst Man Cybern B, 2008, 38: 1207–1220
Chandola V, Banerjee A, Kumar V. Anomaly detection: a survey. ACM Comput Surv, 2009, 41: 1–58
van der Maaten L, Postma E, van den Herik J, et al. Dimensionality reduction: a comparative. J Mach Learn Res, 2009, 10: 66–71
Jain A K, Murty M N, Flynn P J. Data clustering: a review. ACM Comput Surv, 1999, 31: 264–323
Aïmeur E, Brassard G, Gambs S. Quantum clustering algorithms. In: Proceedings of the 24th International Conference on Machine Learning, Corvalis, 2007. 1–8
Giovannetti V, Lloyd S, Maccone L. Quantum random access memory. Phys Rev Lett, 2008, 100: 160501
Farhi E, Goldstone J, Gutmann S, et al. Quantum computation by adiabatic evolution. 2000. ArXiv:quant-ph/0001106v1
Lloyd S, Mohseni M, Rebentrost P. Quantum algorithms for supervised and unsupervised machine learning. 2013. ArXiv:1307.0411
Ng A, Jordan M, Weiss Y. On spectral clustering: analysis and an algorithm. In: Proceedings of the 14th International Conference on Neural Information Processing Systems: Natural and Synthetic, Vancouver, 2001. 849–856
von Luxburg U. A tutorial on spectral clustering. Stat Comput, 2007, 17: 395–416
Daskin A. Quantum spectral clustering through a biased phase estimation algorithm. Turk World Math Soc J Appl Eng Math, 2017, 10: 1
Kerenidis I, Landman J. Quantum spectral clustering. Phys Rev A, 2021, 103: 042415
Grover L K. A fast quantum mechanical algorithm for database search. In: Proceedings of the 28th Annual ACM Symposium on Theory of Computing, Philadelphia, 1996. 212–219
Kitaev A Y. Quantum measurements and the abelian stabilizer problem. 1995. ArXiv:quant-ph/9511026
Berry D W, Ahokas G, Cleve R, et al. Efficient quantum algorithms for simulating sparse hamiltonians. Commun Math Phys, 2007, 270: 359–371
Wilkinson J H. The Algebraic Eigenvalue Problem. Oxford: Oxford University Press, 1988. 570–646
Poulin D, Wocjan P. Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer. Phys Rev Lett, 2009, 103: 220502
Watrous J. Quantum computation: lecture notes. 2006. https://cs.uwaterloo.ca/∼watrous/QC-notes/
Yoder T J, Low G H, Chuang I L. Fixed-point quantum search with an optimal number of queries. Phys Rev Lett, 2014, 113: 210501
Brassard G, Høyer P, Tapp A. Quantum counting. In: Automata, Languages and Programming. Berlin: Springer, 1998. 820–831
Zha H, He X, Ding C, et al. Spectral relaxation for k-means clustering. In: Proceedings of the 14th International Conference on Neural Information Processing Systems: Natural and Synthetic, Vancouver, 2001. 1057–1064
Russell S J, Norvig P. Artificial Intelligence: A Modern Approach. Upper Saddle River: Prentice Hall, 2010. 131–146
Acknowledgements
This work was supported by National Key R&D Program of China (Grant No. 2018YFA0306703).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Li, Q., Huang, Y., Jin, S. et al. Quantum spectral clustering algorithm for unsupervised learning. Sci. China Inf. Sci. 65, 200504 (2022). https://doi.org/10.1007/s11432-022-3492-x
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11432-022-3492-x