- 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

Practical few-atom quantum reservoir computing

Chuanzhou Zhu1,*, Peter J. Ehlers1, Hendra I. Nurdin2, and Daniel Soh1,†

  • *Contact author: chuanzhouzhu@arizona.edu
  • †Contact author: danielsoh@arizona.edu
PDF

Phys. Rev. Research 7, 023290 – Published 20 June, 2025

DOI: https://doi.org/10.1103/wsyq-jyxd

Abstract

Quantum reservoir computing (QRC) harnesses quantum systems to tackle intricate computational problems with exceptional efficiency and minimized energy usage. This paper presents a QRC framework that utilizes a minimalistic quantum reservoir, consisting of only a few two-level atoms within an optical cavity. The system is inherently scalable, as newly added atoms automatically couple with the existing ones through the shared cavity field. We demonstrate that the quantum reservoir outperforms traditional classical reservoir computing in both memory retention and nonlinear data processing through two tasks, namely, the prediction of time-series data using the Mackey-Glass task and the classification of sine-square waveforms. Our results show significant performance improvements with an increasing number of atoms, facilitated by nondestructive, continuous quantum measurements and polynomial regression techniques. These findings confirm the potential of QRC as a practical and efficient solution to addressing complex computational challenges in quantum machine learning.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (61)

  1. K. Fujii and K. Nakajima, Harnessing disordered-ensemble quantum dynamics for machine learning, Phys. Rev. Appl. 8, 024030 (2017).
  2. J. Chen and H. I. Nurdin, Learning nonlinear input–output maps with dissipative quantum systems, Quantum Info. Proc. 18, 198 (2019).
  3. J. Chen, H. I. Nurdin, and N. Yamamoto, Temporal information processing on noisy quantum computers, Phys. Rev. Appl. 14, 024065 (2020).
  4. J. Dudas, B. Carles, E. Plouet, F. A. Mizrahi, J. Grollier, and D. Marković, Quantum reservoir computing implementation on coherently coupled quantum oscillators, npj Quantum Inf. 9, 64 (2023).
  5. L. C. G. Govia, G. J. Ribeill, G. E. Rowlands, H. K. Krovi, and T. A. Ohki, Quantum reservoir computing with a single nonlinear oscillator, Phys. Rev. Res. 3, 013077 (2021).
  6. R. A. Bravo, K. Najafi, X. Gao, and S. F. Yelin, Quantum reservoir computing using arrays of Rydberg atoms, PRX Quantum 3, 030325 (2022).
  7. T. Hülser, F. Köster, K. Lüdge, and L. Jaurigue, Deriving task specific performance from the information processing capacity of a reservoir computer, Nanophotonics 12, 937 (2023).
  8. J. Nokkala, R. Martínez-Peña, G. L. Giorgi, V. Parigi, M. C. Soriano, and R. Zambrini, Gaussian states of continuous-variable quantum systems provide universal and versatile reservoir computing, Commun. Phys. 4, 53 (2021).
  9. R. Martínez-Peña, G. L. Giorgi, J. Nokkala, M. C. Soriano, and R. Zambrini, Dynamical phase transitions in quantum reservoir computing, Phys. Rev. Lett. 127, 100502 (2021).
  10. W. Xia, J. Zou, X. Qiu, and X. Li, The reservoir learning power across quantum many-body localization transition, Front. Phys. 17, 33506 (2022).
  11. P. Mujal, R. Martínez-Peña, G. L. Giorgi, M. C. Soriano, and R. Zambrini, Time-series quantum reservoir computing with weak and projective measurements, npj Quantum Inf. 9, 16 (2023).
  12. D. Fry, A. Deshmukh, S. Y.-C. Chen, V. Rastunkov, and V. Markov, Optimizing quantum noise-induced reservoir computing for nonlinear and chaotic time series prediction, Sci. Rep. 13, 19326 (2023).
  13. W. D. Kalfus, G. J. Ribeill, G. E. Rowlands, H. K. Krovi, T. A. Ohki, and L. C. G. Govia, Hilbert space as a computational resource in reservoir computing, Phys. Rev. Res. 4, 033007 (2022).
  14. T. Yasuda et al., Quantum reservoir computing with repeated measurements on superconducting devices, arXiv:2310.06706.
  15. J. García-Beni, G. L. Giorgi, M. C. Soriano, and R. Zambrini, Scalable photonic platform for real-time quantum reservoir computing, Phys. Rev. Appl. 20, 014051 (2023).
  16. D. Marković, N. Leroux, M. Riou, F. Abreu Araujo, J. Torrejon, D. Querlioz, A. Fukushima, S. Yuasa, J. Trastoy, P. Bortolotti, and J. Grollier, Reservoir computing with the frequency, phase, and amplitude of spin-torque nano-oscillators, Appl. Phys. Lett. 114, 012409 (2019).
  17. J. Lin, Z. Y. Lai, and X. Li, Quantum adiabatic algorithm design using reinforcement learning, Phys. Rev. A 101, 052327 (2020).
  18. L. Innocenti et al., Potential and limitations of quantum extreme learning machines, Commun. Phys. 6, 118 (2023).
  19. S. Ghosh, A. Opala, M. Matuszewski, T. Paterek, and T. C. H. Liew, Quantum reservoir processing, npj Quantum Inf. 5, 35 (2019).
  20. S. Ghosh, T. Krisnanda, T. Paterek, and T. C. H. Liew, Realising and compressing quantum circuits with quantum reservoir computing, Commun. Phys. 4, 105 (2021).
  21. L. Domingo, G. Carlo, and F. Borondo, Optimal quantum reservoir computing for the noisy intermediate-scale quantum era, Phys. Rev. E 106, L043301 (2022).
  22. G. Angelatos, S. A. Khan, and H. E. Türeci, Reservoir computing approach to quantum state measurement, Phys. Rev. X 11, 041062 (2021).
  23. G. Tanaka, T. Yamane, J. B. Héroux, R. Nakane, N. Kanazawa, S. Takeda, H. Numata, D. Nakano, and A. Hirose, Recent advances in physical reservoir computing: A review, Neural Networks 115, 100 (2019).
  24. L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, Information processing using a single dynamical node as complex system, Nat. Commun. 2, 468 (2011).
  25. J. Pathak, B. Hunt, M. Girvan, Z. Lu, and E. Ott, Model-free prediction of large spatiotemporally chaotic systems from data: A reservoir computing approach, Phys. Rev. Lett. 120, 024102 (2018).
  26. X. Chen, F. A. Araujo, M. Riou, J. Torrejon, D. Ravelosona, W. Kang, W. Zhao, J. Grollier, and D. Querlioz, Forecasting the outcome of spintronic experiments with neural ordinary differential equations, Nat. Commun. 13, 1016 (2022).
  27. D. J. Gauthier, E. Bollt, A. Griffith, and W. A. S. Barbosa, Next generation reservoir computing, Nat. Commun. 12, 5564 (2021).
  28. T. Wang, S.-Y. Ma, L. G. Wright, T. Onodera, B. C. Richard, and P. L. McMahon, An optical neural network using less than 1 photon per multiplication, Nat. Commun. 13, 123 (2022).
  29. S.-Y. Ma, T. Wang, J. Laydevant, L. G. Wright, and P. L. McMahon, Quantum-limited stochastic optical neural networks operating at a few quanta per activation, Nat. Commun. 16, 359 (2025).
  30. M. Kornjaca, H.-Y. Hu, C. Zhao, J. Wurtz, P. Weinberg, M. Hamdan, A. Zhdanov, S. H. Cantu, H. Zhou, R. A. Bravo, K. Bagnall, J. I. Basham, J. Campo, A. Choukri, R. DeAngelo, P. Frederick, D. Haines, J. Hammett, N. Hsu, M.-G. Hu et al., Large-scale quantum reservoir learning with an analog quantum computer, arXiv:2407.02553.
  31. F. Hu, S. A. Khan, N. T. Bronn, G. Angelatos, G. E. Rowlands, G. J. Ribeill, and H. E. Türeci, Overcoming the coherence time barrier in quantum machine learning on temporal data, Nat. Commun. 15, 7491 (2024).
  32. K. Kobayashi, K. Fujii, and N. Yamamoto, Feedback-driven quantum reservoir computing for time-series analysis, PRX Quantum 5, 040325 (2024).
  33. O. Ahmed, F. Tennie, and L. Magri, Optimal training of finitely sampled quantum reservoir computers for forecasting of chaotic dynamics, Quantum Mach. Intell. 7, 31 (2025).
  34. H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control (Cambridge University Press, Cambridge, 2009).
  35. L. Bouten, R. van Handel, and M. R. James, An introduction to quantum filtering, SIAM J. Control Optim. 46, 2199 (2007).
  36. H. Wei and Y. V. Nazarov, Statistics of measurement of noncommuting quantum variables: Monitoring and purification of a qubit, Phys. Rev. B 78, 045308 (2008).
  37. R. Ruskov, A. N. Korotkov, and K. Mølmer, Qubit state monitoring by measurement of three complementary observables, Phys. Rev. Lett. 105, 100506 (2010).
  38. S. Hacohen-Gourgy, L. S. Martin, E. Flurin, V. V. Ramasesh, K. B. Whaley, and I. Siddiqi, Quantum dynamics of simultaneously measured non-commuting observables, Nature (London) 538, 491 (2016).
  39. M. A. Ochoa, W. Belzig, and A. Nitzan, Simultaneous weak measurement of non-commuting observables: A generalized Arthurs-Kelly protocol, Sci. Rep. 8, 15781 (2018).
  40. C. A. Fuchs and K. Jacobs, Information-tradeoff relations for finite-strength quantum measurements, Phys. Rev. A 63, 062305 (2001).
  41. S. A. Khan, F. Hu, G. Angelatos, and H. E. Türeci, Physical reservoir computing using finitely-sampled quantum systems, arXiv:2110.13849.
  42. F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, Cavity QED with a Bose–Einstein condensate, Nature (London) 450, 268 (2007).
  43. T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, D. Zueco, T. Hümmer, E. Solano, A. Marx, and R. Gross, Circuit quantum electrodynamics in the ultrastrong-coupling regime, Nat. Phys. 6, 772 (2010).
  44. A. M. Kaufman and K.-K. Ni, Quantum science with optical tweezer arrays of ultracold atoms and molecules, Nat. Phys. 17, 1324 (2021).
  45. M. Ye, Y. Tian, J. Lin, Y. Luo, J. You, J. Hu, W. Zhang, W. Chen, and X. Li, Universal quantum optimization with cold atoms in an optical cavity, Phys. Rev. Lett. 131, 103601 (2023).
  46. N. N. Ledentsov, V. A. Shchukin, M. Grundmann, N. Kirstaedter, J. Böhrer, O. Schmidt, D. Bimberg, V. M. Ustinov, A. Y. Egorov, A. E. Zhukov, P. S. Kop'ev, S. V. Zaitsev, N. Y. Gordeev, Z. I. Alferov, A. I. Borovkov, A. O. Kosogov, S. S. Ruvimov, P. Werner, U. Gösele, and J. Heydenreich, Direct formation of vertically coupled quantum dots in Stranski-Krastanow growth, Phys. Rev. B 54, 8743 (1996).
  47. H. I. Nurdin, Quantum filtering for multiple input multiple output systems driven by arbitrary zero-mean jointly Gaussian input fields, Russ. J. Math. Phys. 21, 386 (2014).
  48. H. M. Wiseman and L. Diósi, Complete parameterization, and invariance, of diffusive quantum trajectories for Markovian open systems, Chem. Phys. 268, 91 (2001).
  49. V. P. Belavkin, Optimal Measurement and Control in Quantum Dynamical Systems, preprint 411 (Institute of Physics, Nicolaus Copernicus University, Torun, 1979).
  50. V. P. Belavkin, Nondemolition measurements, nonlinear filtering, and dynamic programming of quantum stochastic processes, in Modelling and Control of Systems in Engineering, Quantum Mechanics, Economics, and Biosciences, edited by A. Blaquiere (Springer Verlag, New York, 1988), pp. 245–265.
  51. V. P. Belavkin, Continuous non-demolition observation, quantum filtering and optimal estimation, in Quantum Aspects of Optical Communication, Lecture Notes in Physics Vol. 45 (Springer, Berlin, 1991), pp. 151–163.
  52. H. Carmichael, An Open Systems Approach to Quantum Optics (Springer, Berlin, 1993).
  53. F. A. Araujo, M. Riou, J. Torrejon, S. Tsunegi, D. Querlioz, K. Yakushiji, A. Fukushima, H. Kubota, S. Yuasa, M. D. Stiles, and J. Grollier, Role of non-linear data processing on speech recognition task in the framework of reservoir computing, Sci. Rep. 10, 328 (2020).
  54. F. Hu, G. Angelatos, S. A. Khan, M. Vives, E. Türeci, L. Bello, G. E. Rowlands, G. J. Ribeill, and H. E. Türeci, Tackling sampling noise in physical systems for machine learning applications: Fundamental limits and eigentasks, Phys. Rev. X 13, 041020 (2023).
  55. A. Palacios, R. Martínez-Peña, M. C. Soriano, G. L. Giorgi, and R. Zambrini, Role of coherence in many-body quantum reservoir computing, Commun. Phys. 7, 369 (2024).
  56. Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, Optoelectronic reservoir computing, Sci. Rep. 2, 287 (2012).
  57. D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, Parallel photonic information processing at gigabyte per second data rates using transient states, Nat. Commun. 4, 1364 (2013).
  58. C. Zhu, P. J. Ehlers, H. I. Nurdin, and D. Soh, Minimalistic and scalable quantum reservoir computing enhanced with feedback, arXiv:2412.17817.
  59. P. J. Ehlers, H. I. Nurdin, and D. Soh, Improving the performance of echo state networks through state feedback, Neural Networks 184, 107101 (2025).
  60. chuanzhouzhu@arizona.edu.
  61. G. Dion, S. Mejaouri, and J. Sylvestre, Reservoir computing with a single delay-coupled non-linear mechanical oscillator, J. Appl. Phys. 124, 152132 (2018).

Outline

Information

Phys. Rev. Research 7, 023290– Published 20 June, 2025
  • Received 28 January 2025
  • Accepted 4 June 2025
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