- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Quantum-enhanced t-stochastic neighbor embedding for high-dimensional data visualization

Xin Zhang

Chao-Hua Yu*

Gong-De Guo

Song Lin†

  • Digital Fujian Internet-of-Things Laboratory of Environmental Monitoring, Fujian Normal University, Fuzhou 350117, China

  • *Contact author: quantum.ych@gmail.com
  • †Contact author: lins95@fjnu.edu.cn
PDF

Phys. Rev. Research 8, 023265 – Published 8 June, 2026

DOI: https://doi.org/10.1103/w8b9-7zt9

Abstract

t-distributed stochastic neighbor embedding (t-SNE) is a nonlinear dimensionality reduction algorithm that preserves local similarity by matching probability distributions between high- and low-dimensional spaces. Owing to its ability to capture complex manifold structures, t-SNE has become a widely used technique for high-dimensional data visualization. However, it still faces significant computational challenges when dealing with large-scale datasets. In this paper, we present a quantum-enhanced t-SNE algorithm that achieves a quadratic speedup with respect to the number of data samples. Specifically, three unitary operators are constructed using quantum arithmetic and comparators to compute the probability distributions. By integrating them with the Hadamard transform, the gradients of the Kullback-Leibler divergence can be efficiently computed. Moreover, since a complete quantum procedure is provided, our method surpasses the performance of existing quantum variants.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (40)

  1. C. M. Bishop, Pattern Recognition and Machine Learning (Springer, New York, NY, 2006).
  2. H. Li, Z. Xu, G. Taylor, C. Studer, and T. Goldstein, Visualizing the loss landscape of neural nets, in Advances in Neural Information Processing Systems (Curran Associates Inc., Montréal, Canada, 2018).
  3. D. Erhan, A. Courville, Y. Bengio, and P. Vincent, Why does unsupervised pre-training help deep learning? in Proceedings of the 13th International Conference on Artificial Intelligence and Statistics (JMLR Workshop and Conference Proceedings) (JMLR.org, 2010), pp. 201–208.
  4. L. van der Maaten and G. Hinton, Visualizing data using t-SNE, J. Mach. Learn. Res. 9, 2579 (2008).
  5. P. W. Shor, Algorithms for quantum computation: Discrete logarithms and factoring, in Proceedings of the 35th Annual Symposium on Foundations of Computer Science (IEEE, New York, NY, 1994), pp. 124–134.
  6. L. K. Grover, A fast quantum mechanical algorithm for database search, in Proceedings of the 28th Annual ACM Symposium on Theory of Computing (ACM, New York, NY, 1996), pp. 212–219.
  7. A. W. Harrow, A. Hassidim, and S. Lloyd, Quantum algorithm for linear systems of equations, Phys. Rev. Lett. 103, 150502 (2009).
  8. C. H. Yu, F. Gao, and Q. Y. Wen, An improved quantum algorithm for ridge regression, IEEE Trans. Knowl. Data Eng. 33, 858 (2019).
  9. S. Lloyd, M. Mohseni, and P. Rebentrost, Quantum principal component analysis, Nat. Phys. 10, 631 (2014).
  10. I. Cong and L. Duan, Quantum discriminant analysis for dimensionality reduction and classification, New J. Phys. 18, 073011 (2016).
  11. C. H. Yu, F. Gao, S. Lin, and J. B. Wang, Quantum data compression by principal component analysis, Quantum Inf. Process 18, 249 (2019).
  12. W. Feng, G. Guo, S. Lin, and Y. Xu, Quantum Isomap algorithm for manifold learning, Phys. Rev. Appl. 22, 014049 (2024).
  13. H.-L. Liu, L.-C. Wan, C.-H. Yu, S.-J. Pan, S.-J. Qin, F. Gao, and Q.-Y. Wen, A quantum algorithm for solving eigenproblem of the Laplacian matrix of a fully connected weighted graph, Adv. Quantum Technol. 6, 2300031 (2023).
  14. Y. Kawase, K. Mitarai, and K. Fujii, Parametric t-stochastic neighbor embedding with quantum neural network, Phys. Rev. Res. 4, 043199 (2022).
  15. Y. Kawase, K. Mitarai, and K. Fujii, Quantum kernel t-distributed stochastic neighbor embedding, Phys. Rev. Res. 6, 043234 (2024).
  16. Z. Ye, K. Yu, G.-D. Guo, and S. Lin, Quantum self-organizing feature mapping neural network algorithm based on Grover search algorithm, Phys. A: Stat. Mech. Appl. 639, 129690 (2024).
  17. H. Strange and R. Zwiggelaar, Open Problems in Spectral Dimensionality Reduction (Springer, Berlin, Heidelberg, 2014).
  18. L. Van Der Maaten, Barnes-Hut-SNE, in Proceedings of the International Conference on Learning Representations (ICLR, 2013).
  19. L. Van Der Maaten, Accelerating t-SNE using tree-based algorithms, J. Mach. Learn. Res. 15, 3221 (2014).
  20. P. Yianilos, Data structures and algorithms for nearest neighbor search in general metric spaces, in Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SIAM, Philadelphia, PA, 1993), pp. 311–321.
  21. J. Barnes and P. Hut, A hierarchical O(NlogN) force-calculation algorithm, Nature (London) 324, 446 (1986).
  22. J. K. Salmon and M. S. Warren, Skeletons from the treecode closet, J. Comput. Phys. 111, 136 (1994).
  23. M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, et al., Variational quantum algorithms, Nat. Rev. Phys. 3, 625 (2021).
  24. V. Giovannetti, S. Lloyd, and L. Maccone, Quantum random access memory, Phys. Rev. Lett. 100, 160501 (2008).
  25. V. Giovannetti, S. Lloyd, and L. Maccone, Architectures for a quantum random access memory, Phys. Rev. A 78, 052310 (2008).
  26. C. T. Hann, G. Lee, S. Girvin, and L. Jiang, Resilience of quantum random access memory to generic noise, PRX Quantum 2, 020311 (2021).
  27. L. Ruiz-Perez and J. C. Garcia-Escartin, Quantum arithmetic with the quantum Fourier transform, Quantum Inf. Process 16, 152 (2017).
  28. C. P. Shao, Y. Li, and H. B. Li, Quantum algorithm design: Techniques and applications, J. Syst. Sci. Complex. 32, 375 (2019).
  29. G. Brassard, P. Hoyer, M. Mosca, and A. Tapp, Quantum amplitude amplification and estimation, Contemp. Math. 305, 53 (2002).
  30. H. S. Li, P. Fan, H. Y. Xia, and G. L. Long, The circuit design and optimization of quantum multiplier and divider, Sci. China Phys. Mech. Astron. 65, 260311 (2022).
  31. D. Wang, Z. H. Liu, W. N. Zhu, and S. Z. Li, Design of quantum comparator based on extended general Toffoli gates with multiple targets, Comput. Sci. 39, 302 (2012).
  32. I. Kerenidis, J. Landman, A. Luongo, and A. Prakash, q-means: A quantum algorithm for unsupervised machine learning, in Proceedings of the 33rd International Conference on Neural Information Processing Systems (Curran Associates, Red Hook, NY, 2019), pp. 4134–4144.
  33. A. Pavlidis and E. Floratos, Quantum-Fourier-transform-based quantum arithmetic with qudits, Phys. Rev. A 103, 032417 (2021).
  34. R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca, Quantum algorithms revisited, Proc. R. Soc. London Ser. A 454, 339 (1998).
  35. C. H. Yu, H. M. Rao, Y. P. Wu, D. X. Liu, X. P. Liu, and L. C. Wan, Quantum-assisted anomaly detection with multivariate Gaussian distribution, arXiv:2505.02316v2.
  36. S. S. Zhou, T. Loke, J. A. Izaac, and J. B. Wang, Quantum Fourier transform in computational basis, Quantum Inf. Process 16, 82 (2017).
  37. D. Dervovic, M. Herbster, P. Mountney, S. Severini, N. Usher, and L. Wossnig, Quantum linear systems algorithms: A primer, arXiv:1802.08227v1.
  38. J. R. Taylor, An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements (MIT Press, Cambridge, MA, 2022).
  39. I. Kerenidis and A. Prakash, Quantum recommendation systems, in Proceedings of the 8th Innovations in Theoretical Computer Science Conference (Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Wadern, Germany, 2017), pp. 49:1–49:21.
  40. M. Kline, Calculus: An Intuitive and Physical Approach (Courier Corporation, 1998).

Outline

Information

Phys. Rev. Research 8, 023265– Published 8 June, 2026
  • Received 3 September 2025
  • Accepted 6 April 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation