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Theory and interpretability of quantum extreme learning machines: A Pauli transfer matrix approach

Markus Gross* and Hans-Martin Rieser

  • *Contact author: markus.gross@dlr.de
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Phys. Rev. A 114, 042459 – Published 26 October, 2026

DOI: https://doi.org/10.1103/6tm7-3tn4

Abstract

Quantum reservoir computers (QRCs) have emerged as a promising approach to quantum machine learning, since they utilize the natural dynamics of quantum systems for data processing and are simple to train. Here we consider n-qubit quantum extreme learning machines (QELMs) with initial-state encoding and continuous-time reservoir dynamics. Applying the Pauli transfer matrix (PTM) formalism reveals the encoding-generated features and their linear transformation by quantum channels and thereby provides a nonlinear vector autoregression model as an interpretable classical representation of a QELM. We show that the nonlinear information processing capacity of a QELM factorizes into the ability of the encoding to represent a given target function and the decodability of the relevant Pauli features after the effective PTM. Comparing geometric and statistical decodability reveals features that are algebraically hidden yet recoverable through data correlations. We relate the decodability of Pauli features to the spreading of measurement operators under Heisenberg evolution (Krylov growth). Focusing on the unitary channel, we discuss how structured Hamiltonians, when paired with certain observables, can impair model performance, as reflected in a low rank of the readout and the effective PTMs. We trace this effect to Hamiltonian symmetries and derive asymptotic rank estimates for symmetry-resolved observable families. Our decodability analysis disentangles the contributions of encoding, reservoir dynamics, and measurement observables and thus provides interpretable diagnostics that can inform the design of QELMs and reduce reliance on exhaustive hyperparameter search. As a specific application, we consider forecasting nonlinear dynamical systems and show that the readout of a trained QELM closely matches the expression of the underlying flow map at leading orders.

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References (131)

  1. K. Fujii and K. Nakajima, Harnessing disordered-ensemble quantum dynamics for machine learning, Phys. Rev. Appl. 8, 024030 (2017).
  2. A. Sakurai, M. P. Estarellas, W. J. Munro, and K. Nemoto, Quantum extreme reservoir computation utilizing scale-free networks, Phys. Rev. Appl. 17, 064044 (2022).
  3. M. Kornjača, 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.
  4. A. Sakurai, A. Hayashi, W. J. Munro, and K. Nemoto, Simple Hamiltonian dynamics as a powerful resource for image classification, Phys. Rev. A 111, 052432 (2025).
  5. A. De Lorenzis, M. Casado, M. Estarellas, N. Lo Gullo, T. Lux, F. Plastina, A. Riera, and J. Settino, Harnessing quantum extreme learning machines for image classification, Phys. Rev. Appl. 23, 044024 (2025).
  6. J. Chen, H. I. Nurdin, and N. Yamamoto, Temporal information processing on noisy quantum computers, Phys. Rev. Appl. 14, 024065 (2020).
  7. Y. Suzuki, Q. Gao, K. C. Pradel, K. Yasuoka, and N. Yamamoto, Natural quantum reservoir computing for temporal information processing, Sci. Rep. 12, 1353 (2022).
  8. 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).
  9. O. Ahmed, F. Tennie, and L. Magri, Prediction of chaotic dynamics and extreme events: A recurrence-free quantum reservoir computing approach, Phys. Rev. Res. 6, 043082 (2024).
  10. 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).
  11. P. Pfeffer, F. Heyder, and J. Schumacher, Hybrid quantum-classical reservoir computing of thermal convection flow, Phys. Rev. Res. 4, 033176 (2022).
  12. P. Pfeffer, F. Heyder, and J. Schumacher, Reduced-order modeling of two-dimensional turbulent Rayleigh-Bénard flow by hybrid quantum-classical reservoir computing, Phys. Rev. Res. 5, 043242 (2023).
  13. A. Suprano, D. Zia, L. Innocenti, S. Lorenzo, V. Cimini, T. Giordani, I. Palmisano, E. Polino, N. Spagnolo, F. Sciarrino, G. M. Palma, A. Ferraro, and M. Paternostro, Experimental property reconstruction in a photonic quantum extreme learning machine, Phys. Rev. Lett. 132, 160802 (2024).
  14. M. Vetrano, G. Lo Monaco, L. Innocenti, S. Lorenzo, and G. M. Palma, State estimation with quantum extreme learning machines beyond the scrambling time, npj Quantum Inf. 11, 20 (2025).
  15. D. Zia, L. Innocenti, G. Minati, S. Lorenzo, A. Suprano, R. Di Bartolo, N. Spagnolo, T. Giordani, V. Cimini, G. M. Palma, A. Ferraro, F. Sciarrino, and M. Paternostro, Quantum reservoir computing for photonic entanglement witnessing, Sci. Adv. 11, eady7987 (2025).
  16. H. Assil, A. El Allati, and G. L. Giorgi, Entanglement estimation of Werner states with a quantum extreme learning machine, Phys. Rev. A 111, 022412 (2025).
  17. J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nat. Commun. 9, 4812 (2018).
  18. M. Larocca, S. Thanasilp, S. Wang, K. Sharma, J. Biamonte, P. J. Coles, L. Cincio, J. R. McClean, Z. Holmes, and M. Cerezo, Barren plateaus in variational quantum computing, Nat. Rev. Phys. 7, 174 (2025).
  19. L. Domingo, G. Carlo, and F. Borondo, Taking advantage of noise in quantum reservoir computing, Sci. Rep. 13, 8790 (2023).
  20. 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).
  21. A. Sannia, R. Martínez-Peña, M. C. Soriano, G. L. Giorgi, and R. Zambrini, Dissipation as a resource for quantum reservoir computing, Quantum 8, 1291 (2024).
  22. F. Monzani, E. Ricci, L. Nigro, and E. Prati, Non-unital noise in a superconducting quantum computer as a computational resource for reservoir computing, arXiv:2409.07886.
  23. P. Mujal, J. Nokkala, R. Martínez-Peña, G. L. Giorgi, M. C. Soriano, and R. Zambrini, Analytical evidence of nonlinearity in qubits and continuous-variable quantum reservoir computing, J. Phys.: Complexity 2, 045008 (2021).
  24. 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).
  25. L. C. G. Govia, G. J. Ribeill, G. E. Rowlands, and T. A. Ohki, Nonlinear input transformations are ubiquitous in quantum reservoir computing, Neuromorph. Comput. Eng. 2, 014008 (2022).
  26. L. Innocenti, S. Lorenzo, I. Palmisano, A. Ferraro, M. Paternostro, and G. M. Palma, Potential and limitations of quantum extreme learning machines, Commun. Phys. 6, 118 (2023).
  27. J. Chen and H. I. Nurdin, Learning nonlinear input-output maps with dissipative quantum systems, Quantum Info. Proc. 18, 198 (2019).
  28. T. Goto, Q. H. Tran, and K. Nakajima, Universal approximation property of quantum machine learning models in quantum-enhanced feature spaces, Phys. Rev. Lett. 127, 090506 (2021).
  29. R. Martínez-Peña and J.-P. Ortega, Quantum reservoir computing in finite dimensions, Phys. Rev. E 107, 035306 (2023).
  30. F. Monzani and E. Prati, Universality conditions of unified classical and quantum reservoir computing, Neurocomputing 643, 130391 (2025).
  31. L. Gonon and A. Jacquier, Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs, IEEE Trans. Neural Netw. Learn. Syst. 36, 11355 (2025).
  32. M. Schuld, Supervised quantum machine learning models are kernel methods, arXiv:2101.11020.
  33. M. Gross and H.-M. Rieser, Kernel-based optimization of measurement operators for quantum reservoir computers, arXiv:2602.14677.
  34. A. Canatar, E. Peters, C. Pehlevan, S. M. Wild, and R. Shaydulin, Bandwidth enables generalization in quantum kernel models, Transactions on Machine Learning Research (2023).
  35. M. Gross, M. Lange, B. Bujnowski, and H.-M. Rieser, Expressivity vs. generalization in quantum kernel methods, in 33rd European Symposium on Artificial Neural Networks, Computational Intelligence and Machine Learning, ESANN 2025, edited by M. Verleysen (Ciaco - i6doc.com, Bruges, Belgium, 2025), pp. 543–548.
  36. M. Kempkes, A. Ijaz, E. Gil-Fuster, C. Bravo-Prieto, J. Spiegelberg, E. van Nieuwenburg, and V. Dunjko, Double descent in quantum kernel methods, PRX Quantum 7, 010312 (2026).
  37. K. Fujii and K. Nakajima, Quantum reservoir computing: A reservoir approach toward quantum machine learning on near-term quantum devices, in Reservoir Computing: Theory, Physical Implementations, and Applications, Natural Computing Series, edited by K. Nakajima and I. Fischer (Springer, Singapore, 2021), pp. 423–450.
  38. P. Mujal, R. Martínez-Peña, J. Nokkala, J. García-Beni, G. L. Giorgi, M. C. Soriano, and R. Zambrini, Opportunities in quantum reservoir computing and extreme learning machines, Adv. Quantum Technol. 4, 2100027 (2021).
  39. C. M. Wilson, J. S. Otterbach, N. Tezak, R. S. Smith, A. M. Polloreno, P. J. Karalekas, S. Heidel, M. S. Alam, G. E. Crooks, and M. P. da Silva, Quantum kitchen sinks: An algorithm for machine learning on near-term quantum computers, arXiv:1806.08321.
  40. G.-B. Huang, Q.-Y. Zhu, and C.-K. Siew, Extreme learning machine: Theory and applications, Neurocomputing 70, 489 (2006).
  41. A. Rahimi and B. Recht, Random features for large-scale kernel machines, in Advances in Neural Information Processing Systems (Curran Associates, Red Hook, NY, 2007), Vol. 20, pp. 1177–1184.
  42. 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).
  43. L. Domingo, G. Carlo, and F. Borondo, Optimal quantum reservoir computing for the noisy intermediate-scale quantum era, Phys. Rev. E 106, L043301 (2022).
  44. A. Hayashi, A. Sakurai, S. Nishio, W. J. Munro, and K. Nemoto, Impact of the form of weighted networks on the quantum extreme reservoir computation, Phys. Rev. A 108, 042609 (2023).
  45. N. Götting, F. Lohof, and C. Gies, Exploring quantumness in quantum reservoir computing, Phys. Rev. A 108, 052427 (2023).
  46. S. Kobayashi, Q. H. Tran, and K. Nakajima, Coherence influx is indispensable for quantum reservoir computing, arXiv:2409.12693.
  47. S. Čindrak, L. Jaurigue, and K. Lüdge, Engineering quantum reservoirs through Krylov complexity, expressivity, and observability, Phys. Rev. Res. 7, 043190 (2025).
  48. N.-E. Schütte, N. Götting, H. Müntinga, M. List, D. Brunner, and C. Gies, Expressivity limits of quantum reservoir computing, arXiv:2501.15528.
  49. R. Martínez-Peña and J.-P. Ortega, Input-dependence in quantum reservoir computing, Phys. Rev. E 111, 065306 (2025).
  50. A. De Lorenzis, M. P. Casado, N. Lo Gullo, T. Lux, F. Plastina, and A. Riera, Entanglement and classical simulability in quantum extreme learning machines, arXiv:2509.06873.
  51. M. Schuld, R. Sweke, and J. J. Meyer, Effect of data encoding on the expressive power of variational quantum-machine-learning models, Phys. Rev. A 103, 032430 (2021).
  52. S. Shin, Y. S. Teo, and H. Jeong, Exponential data encoding for quantum supervised learning, Phys. Rev. A 107, 012422 (2023).
  53. W. Xiong, G. Facelli, M. Sahebi, O. Agnel, T. Chotibut, S. Thanasilp, and Z. Holmes, On fundamental aspects of quantum extreme learning machines, Quantum Mach. Intell. 7, 20 (2025).
  54. J. Dambre, D. Verstraeten, B. Schrauwen, and S. Massar, Information processing capacity of dynamical systems, Sci. Rep. 2, 514 (2012).
  55. R. Martínez-Peña, J. Nokkala, G. L. Giorgi, R. Zambrini, and M. C. Soriano, Information processing capacity of spin-based quantum reservoir computing systems, Cognit. Comput. 15, 1440 (2023).
  56. Z. Shumaylov, P. Zaika, P. Scholl, G. Kutyniok, L. Horesh, and C.-B. Schönlieb, When is a system discoverable from data? Discovery requires chaos, arXiv:2511.08860.
  57. G. McCaul, J. S. Totero Gongora, W. Otieno, S. Savel'ev, A. Zagoskin, and A. G. Balanov, Minimal quantum reservoirs with Hamiltonian encoding, Chaos 35, 093135 (2025).
  58. M. Schuld and F. Petruccione, Machine Learning with Quantum Computers, 2nd ed., Quantum Science and Technology (Springer, Cham, 2021).
  59. M. C. Caro, Learning quantum processes and Hamiltonians via the Pauli transfer matrix, ACM Trans. Quantum Comput. 5, 1 (2024).
  60. L. Hantzko, L. Binkowski, and S. Gupta, Fast generation of Pauli transfer matrices utilizing tensor product structure, Phys. Scr. 100, 075125 (2025).
  61. A. Angrisani, A. Schmidhuber, M. S. Rudolph, M. Cerezo, Z. Holmes, and H.-Y. Huang, Classically estimating observables of noiseless quantum circuits, Phys. Rev. Lett. 135, 170602 (2025).
  62. A. Angrisani, A. A. Mele, M. S. Rudolph, M. Cerezo, and Z. Holmes, Simulating quantum circuits with arbitrary local noise using Pauli propagation, PRX Quantum 7, 020313 (2026).
  63. M. S. Rudolph, T. Jones, Y. Teng, A. Angrisani, and Z. Holmes, Pauli propagation: A computational framework for simulating quantum systems, PRX Quantum 7, 032001 (2026).
  64. S. Čindrak, B. Donvil, K. Lüdge, and L. Jaurigue, Enhancing the performance of quantum reservoir computing and solving the time-complexity problem by artificial memory restriction, Phys. Rev. Res. 6, 013051 (2024).
  65. J. Steinegger and C. Räth, Predicting three-dimensional chaotic systems with four qubit quantum systems, Sci. Rep. 15, 6201 (2025).
  66. This allows for the existence of a pseudoinverse such that R+R=Iq.
  67. D. J. Gauthier, E. Bollt, A. Griffith, and W. A. S. Barbosa, Next generation reservoir computing, Nat. Commun. 12, 5564 (2021).
  68. H. C. So, K. W. Chan, Y. T. Chan, and K. C. Ho, Linear prediction approach for efficient frequency estimation of multiple real sinusoids: Algorithms and analyses, IEEE Trans. Signal Process. 53, 2290 (2005).
  69. S. V. Vaseghi, Advanced Digital Signal Processing and Noise Reduction, 4th ed. (John Wiley & Sons, Chichester, UK, 2008).
  70. M. Gross, Flow map learning in nonlinear vector autoregressive models: Influence of the feature-library structure on the training error, arXiv:2605.31438.
  71. M. C. Caro, E. Gil-Fuster, J. J. Meyer, J. Eisert, and R. Sweke, Encoding-dependent generalization bounds for parametrized quantum circuits, Quantum 5, 582 (2021).
  72. E. Peters and M. Schuld, Generalization despite overfitting in quantum machine learning models, Quantum 7, 1210 (2023).
  73. 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).
  74. S. Čindrak, K. Lüdge, and L. Jaurigue, From Krylov complexity to observability: Capturing phase space dimension with applications in quantum reservoir computing, Phys. Rev. Res. 7, L042039 (2025).
  75. D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, A universal operator growth hypothesis, Phys. Rev. X 9, 041017 (2019).
  76. A. Nahum, S. Vijay, and J. Haah, Operator spreading in random unitary circuits, Phys. Rev. X 8, 021014 (2018).
  77. E. H. Lieb and D. W. Robinson, The finite group velocity of quantum spin systems, Commun. Math. Phys. 28, 251 (1972).
  78. B. Nachtergaele, Y. Ogata, and R. Sims, Propagation of correlations in quantum lattice systems, J. Stat. Phys. 124, 1 (2006).
  79. C.-F. (Anthony) Chen, A. Lucas, and C. Yin, Speed limits and locality in many-body quantum dynamics, Rep. Prog. Phys. 86, 116001 (2023).
  80. B. J. Mahoney and C. S. Lent, Lieb-Robinson correlation function for the quantum transverse-field Ising model, Phys. Rev. Res. 6, 023286 (2024).
  81. A. Sannia, G. L. Giorgi, and R. Zambrini, Exponential concentration and symmetries in quantum reservoir computing, arXiv:2505.10062.
  82. S. Gopalakrishnan, D. A. Huse, V. Khemani, and R. Vasseur, Hydrodynamics of operator spreading and quasiparticle diffusion in interacting integrable systems, Phys. Rev. B 98, 220303(R) (2018).
  83. A. Avdoshkin and A. Dymarsky, Euclidean operator growth and quantum chaos, Phys. Rev. Res. 2, 043234 (2020).
  84. J. D. Noh, Operator growth in the transverse-field Ising spin chain with integrability-breaking longitudinal field, Phys. Rev. E 104, 034112 (2021).
  85. R. Kaneko and I. Danshita, Dynamics of correlation spreading in low-dimensional transverse-field Ising models, Phys. Rev. A 108, 023301 (2023).
  86. A. Sahu, N. D. Varikuti, B. K. Das, and V. Madhok, Quantifying operator spreading and chaos in Krylov subspaces with quantum state reconstruction, Phys. Rev. B 108, 224306 (2023).
  87. If R contains a t=0 block, then since V(0)=Id2 the corresponding rows of R include ej⊤ for all j∈S (up to basis ordering), hence ej∈row(R). Since R+R is the orthogonal projector onto row(R), (R+R)ej=ej and thus γj2=(R+R)jj=1.
  88. Including t≈0 in the temporal multiplexing scheme improves decodability at early iterations but does not affect the final decodability for sufficiently large L.
  89. By global Hadamard equivalence, the statement flips for Pauli-X as initial operators.
  90. H. Jaeger, Short Term Memory in Echo State Networks, GMD Report 152 (GMD Forschungszentrum Informationstechnik, Sankt Augustin, 2001).
  91. T. Kubota, H. Takahashi, and K. Nakajima, Unifying framework for information processing in stochastically driven dynamical systems, Phys. Rev. Res. 3, 043135 (2021).
  92. In Eq. (4.13), Varuy(u) denotes the variance of y(u) over the distribution of u.
  93. This argument is not fully rigorous, since features like ϕx(i)(ui) have infinite Taylor expansion in ui and can thus correlate locally with raw monomials uim.
  94. W. Xia, J. Zou, X. Qiu, and X. Li, The reservoir learning power across quantum many-body localization transition, Front. Phys. 17, 33506 (2022).
  95. M. N. Ivaki, A. Lazarides, and T. Ala-Nissila, Quantum reservoir computing on random regular graphs, Phys. Rev. A 112, 012622 (2025).
  96. 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).
  97. L. Ljung, System Identification: Theory for the User, 2nd ed. (Prentice Hall PTR, Upper Saddle River, NJ, 1999).
  98. J. C. Sprott, Elegant Chaos: Algebraically Simple Chaotic Flows (World Scientific, Singapore, 2010).
  99. We find that training on increments via ut+1−ut=f(ut) does not significantly improve the forecast horizon, unless one uses small integrator time steps Δt≲0.001.
  100. C. Yang, J. Qiao, H. Han, and L. Wang, Design of polynomial echo state networks for time series prediction, Neurocomputing 290, 148 (2018).
  101. S. L. Brunton, J. L. Proctor, and J. N. Kutz, Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proc. Natl. Acad. Sci. USA 113, 3932 (2016).
  102. T.-C. Chen, S. G. Penny, T. A. Smith, and J. A. Platt, ‘Next generation’ reservoir computing: An empirical data-driven expression of dynamical equations in time-stepping form, arXiv:2201.05193.
  103. Y. Zhang, E. Roque dos Santos, H. Zhang, and S. P. Cornelius, How more data can hurt: Instability and regularization in next-generation reservoir computing, Chaos 35, 073102 (2025).
  104. Moreover, nonorthogonal function libraries are typically highly redundant, reflected in large condition numbers of the corresponding design matrices [Eqs. (3.16) and (3.17)].
  105. Deviations are small for N<q: in practice, for n=D=3 (thus q=27) and monomials of order r=3 (thus N=19 excluding the constant term), we get rank(K)=16.
  106. R. Sweke, E. Recio-Armengol, S. Jerbi, E. Gil-Fuster, B. Fuller, J. Eisert, and J. J. Meyer, Potential and limitations of random Fourier features for dequantizing quantum machine learning, Quantum 9, 1640 (2025).
  107. F. J. Schreiber, J. Eisert, and J. J. Meyer, Classical surrogates for quantum learning models, Phys. Rev. Lett. 131, 100803 (2023).
  108. C. Mingard, J. Pointing, C. London, Y. Nam, and A. A. Louis, Exploiting the equivalence between quantum neural networks and perceptrons, arXiv:2407.04371.
  109. S. Masot-Llima, E. Gil-Fuster, C. Bravo-Prieto, J. Eisert, and T. Guaita, Prospects for quantum advantage in machine learning from the representability of functions, arXiv:2512.15661.
  110. S. L. Brunton and J. N. Kutz, Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control (Cambridge University Press, Cambridge, UK, 2019).
  111. E. Gil-Fuster, J. Eisert, and C. Bravo-Prieto, Understanding quantum machine learning also requires rethinking generalization, Nat. Commun. 15, 2277 (2024).
  112. S. Han, L. Jia, and L. Guo, Multiple embeddings for quantum machine learning, arXiv:2503.22758.
  113. J. Bowles, S. Ahmed, and M. Schuld, Better than classical? The subtle art of benchmarking quantum machine learning models, arXiv:2403.07059.
  114. J. Schnabel and M. Roth, Quantum kernel methods under scrutiny: A benchmarking study, Quantum Mach. Intell. 7, 58 (2025).
  115. D. Basilewitsch, J. F. Bravo, C. Tutschku, and F. Struckmeier, Quantum neural networks in practice: A comparative study with classical models from standard data sets to industrial images, Quantum Mach. Intell. 7, 110 (2025).
  116. T. Rohe, D. Schuman, J. Nüßlein, L. Sünkel, J. Stein, and C. Linnhoff-Popien, The questionable influence of entanglement in quantum optimisation algorithms, arXiv:2407.17204.
  117. H. Gundlach, H. Kukina, J. Lynch, and N. Thompson, Quantum deep learning still needs a quantum leap, in Proceedings of the 43rd ICML, Vol. 306 (PMLR, 2026), pp. 170357–170374.
  118. T. Fellner, D. A. Kreplin, S. Tovey, and C. Holm, Quantum vs. classical: A comprehensive benchmark study for predicting time series with variational quantum machine learning, Mach. Learn.: Sci. Technol. 7, 010501 (2026).
  119. D. Freinberger and P. Moser, The role of quantum in hybrid quantum-classical neural networks: A realistic assessment, arXiv:2601.04732.
  120. F. J. Gil Vidal and D. O. Theis, Input redundancy for parameterized quantum circuits, Front. Phys. 8, 297 (2020).
  121. A. Pérez-Salinas, A. Cervera-Lierta, E. Gil-Fuster, and J. I. Latorre, Data re-uploading for a universal quantum classifier, Quantum 4, 226 (2020).
  122. J. Kübler, S. Buchholz, and B. Schölkopf, The inductive bias of quantum kernels, in Advances in Neural Information Processing Systems, edited by M. Ranzato, A. Beygelzimer, Y. Dauphin, P. S. Liang, and J. W. Vaughan (Curran Associates, Red Hook, NY, 2021), Vol. 34, pp. 12661–12673.
  123. For the real product encodings considered here, any Pauli string containing a Y factor has zero pre-reservoir expectation, leaving at most ∼3n features.
  124. https://qci.dlr.de/NeMoQC.
  125. M. B. Ruskai, S. Szarek, and E. Werner, An analysis of completely-positive trace-preserving maps on M2, Lin. Algebr. Appl. 347, 159 (2002).
  126. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th ed. (Cambridge University Press, Cambridge, UK, 2010).
  127. M. M. Wilde, Quantum Information Theory, 2nd ed. (Cambridge University Press, Cambridge, UK, 2017).
  128. J. Watrous, The Theory of Quantum Information (Cambridge University Press, Cambridge, UK, 2018).
  129. T. Kubota, Y. Suzuki, S. Kobayashi, Q. H. Tran, N. Yamamoto, and K. Nakajima, Temporal information processing induced by quantum noise, Phys. Rev. Res. 5, 023057 (2023).
  130. T. Prosen and I. Pižorn, Operator space entanglement entropy in a transverse Ising chain, Phys. Rev. A 76, 032316 (2007).
  131. G. B. Mbeng, A. Russomanno, and G. E. Santoro, The quantum Ising chain for beginners, SciPost Phys. Lect. Notes 82 (2024).

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Phys. Rev. A 114, 042459– Published 26 October, 2026
  • Received 8 April 2026
  • Accepted 5 October 2026
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