Abstract
We study quantum imaging by applying the resolvable expressive capacity (REC) formalism developed for physical neural networks (PNNs). In this paradigm of quantum learning, the imaging system functions as a physical learning device that maps input parameters to measurable features, while complex practical tasks are handled by training only the output weights, enabled by the systematic identification of well-estimated features (eigentasks) and their corresponding sample thresholds. Using this framework, we analyze both direct imaging and superresolution strategies for compact sources, defined as sources with sizes bounded below the Rayleigh limit. In particular, we introduce the orthogonalized SPADE method—a nontrivial generalization of existing superresolution techniq…
Abstract
We study quantum imaging by applying the resolvable expressive capacity (REC) formalism developed for physical neural networks (PNNs). In this paradigm of quantum learning, the imaging system functions as a physical learning device that maps input parameters to measurable features, while complex practical tasks are handled by training only the output weights, enabled by the systematic identification of well-estimated features (eigentasks) and their corresponding sample thresholds. Using this framework, we analyze both direct imaging and superresolution strategies for compact sources, defined as sources with sizes bounded below the Rayleigh limit. In particular, we introduce the orthogonalized SPADE method—a nontrivial generalization of existing superresolution techniques—that achieves superior performance when multiple compact sources are closely spaced. This method relaxes the earlier superresolution studies’ strong assumption that the entire source must lie within the Rayleigh limit, marking an important step toward developing more general and practically applicable approaches. Using the example of face recognition, which involve complex structured sources, we demonstrate the superior performance of our orthogonalized SPADE method and highlight key advantages of the quantum learning approach—its ability to tackle complex imaging tasks and enhance performance by selectively extracting well-estimated features.
Data availability
No data sets were generated or analyzed during the current study.
Code availability
All codes used in this paper have been deposited in GitHub at https://github.com/ykwang-phys/quantum-learning-imaging.
References
Tsang, M., Nair, R. & Lu, X.-M. Quantum theory of superresolution for two incoherent optical point sources. Phys. Rev. X 6, 031033 (2016).
Pirandola, S., Bardhan, B. R., Gehring, T., Weedbrook, C. & Lloyd, S. Advances in photonic quantum sensing. Nat. Photonics 12, 724–733 (2018).
Sorelli, G., Gessner, M., Walschaers, M. & Treps, N. Optimal observables and estimators for practical superresolution imaging. Phys. Rev. Lett. 127, 123604 (2021).
Grace, M. R., Dutton, Z., Ashok, A. & Guha, S. Approaching quantum-limited imaging resolution without prior knowledge of the object location. J. Optical Soc. Am. A 37, 1288–1299 (2020).
Tsang, M. Quantum limit to subdiffraction incoherent optical imaging. Phys. Rev. A 99, 012305 (2019).
Tsang, M. Subdiffraction incoherent optical imaging via spatial-mode demultiplexing. N. J. Phys. 19, 023054 (2017).
Zhou, S. & Jiang, L. Modern description of Rayleigh’s criterion. Phys. Rev. A 99, 013808 (2019).
Wang, Y., Zhang, Y. & Lorenz, V. O. Superresolution in interferometric imaging of strong thermal sources. Phys. Rev. A 104, 022613 (2021).
Nair, R. & Tsang, M. Far-field superresolution of thermal electromagnetic sources at the quantum limit. Phys. Rev. Lett. 117, 190801 (2016).
Lupo, C. & Pirandola, S. Ultimate precision bound of quantum and subwavelength imaging. Phys. Rev. Lett. 117, 190802 (2016).
Napoli, C., Piano, S., Leach, R., Adesso, G. & Tufarelli, T. Towards superresolution surface metrology: quantum estimation of angular and axial separations. Phys. Rev. Lett. 122, 140505 (2019).
Yu, Z. & Prasad, S. Quantum limited superresolution of an incoherent source pair in three dimensions. Phys. Rev. Lett. 121, 180504 (2018).
Ang, S. Z., Nair, R. & Tsang, M. Quantum limit for two-dimensional resolution of two incoherent optical point sources. Phys. Rev. A 95, 063847 (2017).
Yang, F., Tashchilina, A., Moiseev, E. S., Simon, C. & Lvovsky, A. I. Farfield linear optical superresolution via heterodyne detection in a higher-order local oscillator mode. Optica 3, 1148–1152 (2016).
Tang, Z. S., Durak, K. & Ling, A. Fault-tolerant and finite-error localization for point emitters within the diffraction limit. Opt. Express 24, 22004–22012 (2016).
Paúr, M., Stoklasa, B., Hradil, Z., Sánchez-Soto, L. L. & Rehacek, J. Achieving the ultimate optical resolution. Optica 3, 1144–1147 (2016).
Tham, W.-K., Ferretti, H. & Steinberg, A. M. Beating Rayleigh’s curse by imaging using phase information. Phys. Rev. Lett. 118, 070801 (2017).
Parniak, M. et al. Beating the Rayleigh limit using two-photon interference. Phys. Rev. Lett. 121, 250503 (2018).
Santamaria, L., Sgobba, F. & Lupo, C. Single-photon sub-Rayleigh precision measurements of a pair of incoherent sources of unequal intensity. Opt. Quantum 2, 46–56 (2024).
Tan, X.-J. et al. Quantum-inspired superresolution for incoherent imaging. Optica 10, 1189–1194 (2023).
Rouvière, C. et al. Ultra-sensitive separation estimation of optical sources. Optica 11, 166–170 (2024).
Zanforlin, U. et al. Optical quantum super-resolution imaging and hypothesis testing. Nat. Commun. 13, 5373 (2022).
Huang, Z. & Lupo, C. Quantum hypothesis testing for exoplanet detection. Phys. Rev. Lett. 127, 130502 (2021).
Zhang, H., Kumar, S. & Huang, Y.-P. Super-resolution optical classifier with high photon efficiency. Opt. Lett. 45, 4968–4971 (2020).
Lu, X.-M., Krovi, H., Nair, R., Guha, S. & Shapiro, J. H. Quantum-optimal detection of one-versus-two incoherent optical sources with arbitrary separation. npj Quantum Inf. 4, 64 (2018).
Grace, M. R. & Guha, S. Identifying objects at the quantum limit for superresolution imaging. Phys. Rev. Lett. 129, 180502 (2022).
Boyd, S. & Chua, L. Fading memory and the problem of approximating nonlinear operators with Volterra series. IEEE Trans. Comput. -Aided Des. Integr. Circuits Syst. 32, 1150 (1985).
Tanaka, G. et al. Recent advances in physical reservoir computing: a review. Neural Netw. 115, 100 (2019).
Mujal, P. et al. Opportunities in Quantum Reservoir Computing and extreme learning machines. Adv. Quantum Technol. 4, 2100027 (2021).
Wilson, C. M. et al. Quantum kitchen sinks: an algorithm for machine learning on near-term quantum computers. Preprint at arXiv https://doi.org/10.48550/arXiv.1806.08321 (2018). 1.
García-Beni, J., Giorgi, G. L., Soriano, M. C. & Zambrini, R. Scalable photonic platform for real-time quantum reservoir computing. Phys. Rev. Appl. 20, 014051 (2023).
Havlíček, V. et al. Supervised learning with quantum-enhanced feature spaces. Nat. (Lond.) 567, 209 (2019).
Rowlands, G. E. et al. Reservoir computing with superconducting electronics. Preprint at arXiv:2103.02522 (2021). 1.
Lin, X. et al. All-optical machine learning using diffractive deep neural networks. Science 361, 1004 (2018).
Pai, S. et al. Experimentally realized in situ backpropagation for deep learning in photonic neural networks. Science 380, 398 (2023).
Dambre, J., Verstraeten, D., Schrauwen, B. & Massar, S. Information processing capacity of dynamical systems. Sci. Rep. 2, 514 (2012).
Sheldon, F. C., Kolchinsky, A. & Caravelli, F. Computational capacity of LRC, memristive and hybrid reservoirs. Phys. Rev. E 106, 045310 (2022).
Schuld, M., Sweke, R. & Meyer, J. J. Effect of data encoding on the expressive power of variational quantum-machine-learning models. Phys. Rev. A 103, 032430 (2021).
Wu, Y., Yao, J., Zhang, P. & Zhai, H. Expressivity of quantum neural networks. Phys. Rev. Res. 3, L032049 (2021).
Hu, F. et al. Tackling sampling noise in physical systems for machine learning applications: fundamental limits and eigentasks. Phys. Rev. X 13, 041020 (2023).
Ozer, I., Grace, M. R. & Guha, S. Reconfigurable spatial-mode sorter for super-resolution imaging. In Conference on Lasers and Electro-Optics (CLEO), 1–2 (IEEE, 2022). 1.
Lavery, M. P. et al. Refractive elements for the measurement of the orbital angular momentum of a single photon. Opt. express 20, 2110–2115 (2012).
Beijersbergen, M. W., Allen, L., Van der Veen, H. & Woerdman, J. Astigmatic laser mode converters and transfer of orbital angular momentum. Opt. Commun. 96, 123–132 (1993).
Ionicioiu, R. Sorting quantum systems efficiently. Sci. Rep. 6, 25356 (2016).
Zhou, Y. et al. Sorting photons by radial quantum number. Phys. Rev. Lett. 119, 263602 (2017).
Zhou, Y. et al. Hermite–Gaussian mode sorter. Opt. Lett. 43, 5263–5266 (2018).
O’Donnell, R. & Wright, J. Efficient Quantum Tomography. In Proceedings of the forty-eighth annual ACM symposium on Theory of Computing, 899–912 (2016). 1.
Haah, J., Harrow, A. W., Ji, Z., Wu, X. & Yu, N. Sample-optimal tomography of quantum states. In Proceedings of the forty-eighth annual ACM symposium on Theory of Computing, 913–925 (2016). 1.
Metger, T., Poremba, A., Sinha, M. & Yuen, H. Pseudorandom unitaries with non-adaptive security. arXiv preprint arXiv:2402.14803 (2024). 1.
Ma, F. & Huang, H.-Y. How to construct random unitaries. arXiv preprint arXiv:2410.10116 (2024). 1.
Schuster, T., Haferkamp, J. & Huang, H.-Y. Random unitaries in extremely low depth. Science 389, 92 (2025).
Acknowledgements
We would like to thank Hakan E. Tureci for helpful discussion. Y.W. and S.Z. acknowledge funding provided by Perimeter Institute for Theoretical Physics, a research institute supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities. Y.W. also acknowledges funding from the Canada First Research Excellence Fund. L.J. acknowledges support from the ARO(W911NF-23-1-0077), ARO MURI (W911NF-21-1-0325), AFOSR MURI (FA9550-19-1-0399, FA9550-21-1-0209, FA9550-23-1-0338), DARPA (HR0011-24-9-0359, HR0011-24-9-0361), NSF (OMA-1936118, ERC-1941583, OMA-2137642, OSI-2326767, CCF-2312755), NTT Research, Packard Foundation (2020-71479), and the Marshall and Arlene Bennett Family Research Program. J.L. is supported in part by the University of Pittsburgh, School of Computing and Information, Department of Computer Science, Pitt Cyber, Pitt Momentum fund, PQI Community Collaboration Awards, John C. Mascaro Faculty Scholar in Sustainability, Thinking Machines Lab, Cisco Research, funding from IBM Quantum through the Chicago Quantum Exchange, and AFOSR MURI (FA9550-21-1-0209). C.O. was supported by the NRF Grants (No. RS-2024-00431768 and No. RS-2025-00515456) funded by the Korean government (MSIT) and IITP grants funded by the Korean government (MSIT) (No. IITP-2025-RS-2025-02283189 and IITP-2025-RS-2025-02263264).
Author information
Authors and Affiliations
Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada
Yunkai Wang & Sisi Zhou 1.
Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada
Yunkai Wang & Sisi Zhou 1.
Institute for Quantum Computing, University of Waterloo, Waterloo, ON, Canada
Yunkai Wang & Sisi Zhou 1.
Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea
Changhun Oh 1.
Department of Computer Science, The University of Pittsburgh, Pittsburgh, PA, USA
Junyu Liu 1.
Pritzker School of Molecular Engineering, The University of Chicago, Chicago, IL, USA
Junyu Liu & Liang Jiang 1.
Department of Physics and Astronomy, University of Waterloo, Waterloo, ON, Canada
Sisi Zhou
Authors
- Yunkai Wang
- Changhun Oh
- Junyu Liu
- Liang Jiang
- Sisi Zhou
Contributions
Y.W. carried out the analytical calculation and the numerical simulation. L.J. conceived the project. S.Z., J.L., and L.J. supervised the project. Y.W., C.O., J.L., L.J., and S.Z. contributed to the development of ideas and the writing of the manuscript.
Corresponding authors
Correspondence to Yunkai Wang, Liang Jiang or Sisi Zhou.
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature Communications thanks Giacomo Sorelli and the other, anonymous, reviewers for their contribution to the peer review of this work. A peer review file is available.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Wang, Y., Oh, C., Liu, J. et al. Advancing quantum imaging through learning theory. Nat Commun (2025). https://doi.org/10.1038/s41467-025-67884-1
Received: 02 February 2025
Accepted: 11 December 2025
Published: 27 December 2025
DOI: https://doi.org/10.1038/s41467-025-67884-1