Pith. sign in

REVIEW 4 major objections 5 minor 71 references

A five-qubit reservoir, read at two different times, forecasts low-dimensional chaos for 100-200 valid-prediction steps and about 13 Lyapunov times.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Temporal multiplexing with two quantum evolution times raises valid prediction time in a five-qubit hybrid reservoir computer and yields matching optimal parameter regions for two chaotic systems.

T0 review reviewed 2026-08-05 challenge →

load-bearing objection Temporal multiplexing clearly helps their hybrid QRC, but the 'competitive' claim is not backed by any baseline comparison. the 4 major comments →

arxiv 2509.04006 v1 pith:HIG3WF4Z submitted 2025-09-04 quant-ph cond-mat.dis-nnnlin.CDphysics.comp-ph

Forecasting Low-Dimensional Turbulence via Multi-Dimensional Hybrid Quantum Reservoir Computing

L. Salatino , L. Mariani , A. Giordano , F. D'Amore , C. Mastroianni , L. Pontieri , A. Vinci , C. Gencarelli
show 4 more authors
L. Primavera F. Plastina J. Settino F. Carbone
This is my paper
classification quant-ph cond-mat.dis-nnnlin.CDphysics.comp-ph
keywords quantum reservoir computingtime series forecastingchaos predictiontemporal multiplexingNavier-Stokes truncationLorenz-63transverse-field Ising modelvalid prediction time
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to show that a hybrid quantum-classical reservoir computer can forecast chaotic multivariate dynamics on two fluid-related benchmarks: a five-mode Galerkin truncation of the two-dimensional Navier-Stokes equations and the Lorenz-63 system. Its central lever is temporal multiplexing: at each input step the five-qubit spin system evolves for two distinct time intervals, and measurements from both evolutions are concatenated before a classical memory layer and a linear readout produce the forecast. On the Navier-Stokes testbed, the Valid Prediction Time with error threshold 0.3 reaches about 159 at forcing F=33 and about 196 at F=28.718, while a single evolution time saturates near 10; Lorenz-63 is predicted for about 13 Lyapunov times. The authors also report that the optimal Hamiltonian parameters are the same for both systems, indicating that the encoding scheme may transfer between tasks without re-optimization. If these results hold, low-dimensional quantum reservoirs plus classical memory offer a practical route to nonlinear time-series forecasting in settings where classical methods struggle.

Core claim

The central claim: a five-qubit transverse-field Ising reservoir, with the input modulating local fields, forecasts low-dimensional turbulent dynamics when its state is read at two different evolution times. With this temporal multiplexing, the Valid Prediction Time (the horizon over which normalized error stays below 0.3) reaches about 159 at F=33 and 196 at F=28.718 for the Navier-Stokes truncation, and about 13 Lyapunov times for Lorenz-63; a single readout saturates near 10. The same optimal parameter region works for both systems, suggesting the encoding transfers across forecasting tasks.

What carries the argument

The engine is a five-qubit transverse-field Ising Hamiltonian, H_k = sum J_ij sigma^x_i sigma^x_j + h sum sigma^z_i + sum h_i(k) sigma^x_i, in which the input vector at step k modulates the local longitudinal fields h_i(k). At each step the state evolves under H_k for two different intervals Delta t_1 and Delta t_2; the expectation values of single-spin components and two-spin correlations from both evolutions are concatenated into one measurement vector. A classical reservoir state, updated by a cyclic permutation and by this measurement vector, supplies memory, and a ridge-regression readout maps it to the forecast. Temporal multiplexing is the load-bearing mechanism: it is what turns a ti

Load-bearing premise

The load-bearing assumption is that staying within the normalized error 0.3 for as long as possible (VPT) is the right measure of forecast quality, and that the five-mode Galerkin truncation is a representative stand-in for low-dimensional turbulence; if either choice is unrepresentative, the reported prediction horizons do not imply what the conclusions claim.

What would settle it

Replace the quantum evolution with a classical reservoir of comparable feature count (for example an echo-state network with the same number of effective units and the same temporal-multiplexing trick) on the same data: if the classical version matches or exceeds the reported VPT, the quantum encoding is not doing the work. Alternatively, repeat the same scan on a 10- or 20-mode Galerkin truncation of the Navier-Stokes system: if the VPT collapses, the result is an artifact of the five-mode benchmark.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Temporal multiplexing is decisive: with one evolution time the Valid Prediction Time peaks near 10 steps, while two distinct times raise it by about two orders of magnitude; equal times behave like a single time.
  • The same optimal region in the (J,h) parameter space works for both benchmarks, so the quantum encoding may not need per-task re-optimization for similar chaotic systems.
  • Small quantum hardware is sufficient for the tested tasks: five qubits plus a classical memory and linear readout forecast a five-mode turbulent system and Lorenz-63 for the reported horizons.
  • The forecasts are robust to random coupling realizations near the optimum, with relative VPT errors that are small when J is small and grow when J becomes comparable to the input scale.
  • For Lorenz-63, the useful region of evolution times is much narrower than for Navier-Stokes, so the choice of multiplexing times becomes task-specific even though the Hamiltonian parameters transfer.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The shared optimal parameter region suggests a scaling principle the paper does not state: the input-dependent field should dominate both the static transverse field and the random couplings, so the reservoir's internal dynamics amplify rather than overwhelm the input. A direct test would be varying input amplitude at fixed J and h.
  • Because the benchmark is a five-mode truncation, the method's relevance to genuinely multiscale 2D turbulence is untested; a higher-order truncation or a spatiotemporal system would show whether the forecasting horizon survives additional degrees of freedom.
  • No equally sized classical reservoir baseline is reported, so the specific quantum contribution, as opposed to the classical memory layer and temporal multiplexing, is not yet isolated.
  • The VPT maxima often lie at the edge of the scanned grid, so the reported values may be limited by the scan range rather than the true predictive ceiling of the reservoir.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes a hybrid quantum-classical reservoir computer for multivariate time-series forecasting. A five-qubit transverse-field Ising Hamiltonian with an input-dependent field is evolved for two different times, and the concatenated measurements are fed through a classical memory layer (Eq. 5) before a linear readout is trained. The method is tested on a five-mode Galerkin truncation of the 2D Navier-Stokes equations (Eq. 10) at two forcing values and on the Lorenz-63 system. Performance is measured by the Valid Prediction Time (Eq. 11). The authors report that temporal multiplexing increases VPT by about two orders of magnitude (Fig. 4), that optimal Hamiltonian parameters cluster at small coupling J and moderate transverse field h (Figs. 5, 8, 11), and that the method achieves VPT values around 159 and 195 time steps for the NS truncation and about 13 Lyapunov times for Lorenz-63.

Significance. If the reported results are reproducible, the temporal-multiplexing mechanism is a clear and useful improvement over single-time quantum reservoir encoding, and the observation that the optimal parameter region is similar for two different chaotic benchmarks is interesting. The paper includes systematic parameter scans and, in the appendices, relative-error heatmaps, which are strengths. However, the central claim of 'competitive prediction performance' is not established because the manuscript provides no quantitative comparison against a classical reservoir computer, a standard echo state network, or the literature values cited in Appendix B. The absolute VPT numbers have no reference point. The paper is therefore a promising proof-of-concept, but the competitiveness claim needs additional baseline evidence before it can be accepted.

major comments (4)
  1. [Sec. IV and Appendix B] The conclusion states that the QRC algorithm achieves 'competitive prediction performance in terms of the Valid Prediction Time,' but no baseline is evaluated. Fig. 4 compares QRC with and without temporal multiplexing, which supports the multiplexing benefit, not competitiveness. Appendix B claims predictions are 'compared to those reported in the literature [45,72]' but no numerical VPT values from those references are given. This is load-bearing because the central claim is an AND: capturing nonlinear dynamics AND achieving competitive performance. Please add a classical reservoir computing baseline (e.g., echo state network) on the same tasks with the same VPT metric, and/or explicitly tabulate the VPT values from refs. [45,72] and explain how the comparison is made.
  2. [Sec. II, Eq. (5)-(6)] The training and evaluation protocol is underspecified. The grid search is described as scanning gamma, J, h, Delta t1, and Delta t2, but the selected value of gamma (memory retention) is never reported, nor are the ridge parameter lambda, the number of training steps Ntr, or the number of realizations used for the F=33 heatmap. Since gamma directly controls the classical memory that is central to the method, omitting it prevents reproducibility. In addition, the manuscript does not state whether predictions are generated in closed loop (predicted outputs fed back as inputs) or open loop; Eq. (6) is only a one-step readout. Please specify the recursive prediction procedure and report the hyperparameters actually used.
  3. [Sec. III.C, Fig. 5] The main F=33 heatmap in the (J,h) plane is reported without error bars or standard deviation, even though the Hamiltonian couplings J_ij are randomly sampled and the text emphasizes variability across realizations. The appendices provide relative-error heatmaps for the time parameters and for F=28.718, but not for the F=33 (J,h) plane. Given that the conclusion about optimal J and h regions relies on this figure, an analogous error analysis should be added. Without it, one cannot distinguish parameter regions with high mean but high variance.
  4. [Appendix B, Figs. 10 and 12] There is a numerical inconsistency in the Lorenz VPT values. The text and Fig. 12 caption say the best prediction extends to about 13 Lyapunov times, but the heatmap in Fig. 10 (Top), which is described as the average over 30 realizations, shows a maximum of 10.84 LT. Clarify whether 13 LT is a single-realization maximum and 10.84 LT is the mean, or whether the normalization differs. This matters because the headline 'roughly 13 Lyapunov times' is cited as evidence of performance.
minor comments (5)
  1. [Eq. (10)] The sentence 'u*_k = u_k, due to the reality condition u_k in R^N' should read 'u_k in R' (or 'u_k is real'); the subscript k already labels the mode. The notation R^N is confusing here.
  2. [Fig. 4] The y-axis label 'VPT ± VPT' is ambiguous. It should be 'VPT ± sigma_VPT' or similar, and the definition of the error bar should be given in the caption.
  3. [Sec. III.A and Eq. (7)] The forcing F is called the 'kinetic Reynolds number' in Sec. III.B, but F is an external forcing term, not a Reynolds number. The terminology is confusing and should be changed to 'forcing amplitude' or 'forcing parameter.'
  4. [References] Several references are duplicated: [4] duplicates [2], [27] duplicates [22], and [57] duplicates [53]. These should be consolidated.
  5. [Fig. 1 and Eq. (3)] The schematic uses m_k^l, while Eq. (3) defines m_k^{(l)}. Unify the notation for the measurement vectors.

Circularity Check

0 steps flagged

No circular derivation: forecasts are evaluated on held-out future data; the main caveats are post-selected hyperparameters and an unquantified 'competitive' claim, neither of which makes the derivation circular.

full rationale

The paper's chain is empirical rather than deductive: Eq. (6) trains a linear readout by ridge regression on reservoir states from the training segment, and the VPT of Eq. (11) is computed on later, held-out trajectory points. There is no equation in which an input is defined by the output or in which a fitted parameter is renamed a prediction. The hyperparameter scan (Sec. II: 'the optimal configuration was selected based on the highest average Valid Prediction Time') does select J, h, Δt1, Δt2 on the same benchmark used for reporting VPT; this is a post-selection/bias concern that inflates the absolute numbers, but the reported forecasts are not forced by construction and the reservoir dynamics is a fixed quantum evolution plus classical memory. The citations to the authors' own prior work ([50] for the memory-augmented scheme, [52,60,64] for the NS truncation) are background/benchmark sources whose content is restated in the paper; no load-bearing claim is justified solely by a self-citation. The absence of a classical baseline means 'competitive prediction performance' (Conclusions) is not demonstrated, but lack of comparison is an evidence/validity issue, not circularity. The Lorenz-63 benchmark is external and the NS equations are written out explicitly, so the evaluation is self-contained. Overall: no circular step; score 2 reflects the minor non-load-bearing self-citation for the architecture.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

All performance-critical quantities (J, h, Delta t1, Delta t2, gamma, epsilon) are either fitted by grid search or chosen by hand; the paper is an empirical demonstration, not a first-principles derivation. No new physical entities are introduced.

free parameters (5)
  • Coupling strength J = optimal small values (0.001 to 0.01)
    Scanned over grid; VPT maximized for small J.
  • Transverse magnetic field h = optimal around h = 1
    Scanned; larger h suppresses input influence and degrades VPT.
  • Evolution times Delta t1, Delta t2 = Delta t1=2, Delta t2=1 for main scan
    Distinct times are required for multiplexing; optimal times vary with system.
  • Memory retention parameter gamma = not reported
    Scanned in grid search, but the selected value is not disclosed.
  • VPT error threshold epsilon = 0.3
    Hand-chosen; no sensitivity analysis is provided.
axioms (5)
  • domain assumption The reservoir state update rule rk = gamma S rk-1 + B mk (Eq. 5) with a linear readout is sufficient to learn the target dynamics.
    Inherited from reservoir computing paradigm; no proof of universality for these systems is given.
  • domain assumption The five-mode Galerkin truncation of the Navier-Stokes equations (Eq. 10) faithfully represents low-dimensional turbulent behavior.
    Section III.B; the truncation is used as the ground truth without comparison to higher-order truncations.
  • domain assumption Expectation values of single-qubit and two-qubit Pauli operators (Eq. 4) provide an informative feature set for forecasting.
    Section II; the choice of observables is fixed and not systematically varied.
  • ad hoc to paper The VPT threshold epsilon=0.3 (Eq. 11) is a valid measure of prediction quality.
    The threshold is chosen by the authors and no justification or sensitivity analysis is provided.
  • domain assumption Input encoding with beta_j=1 and C_ij=delta_ij, normalized to [0,1], preserves the information needed for forecasting.
    Section II; the encoding is an arbitrary choice not tested against alternatives.

reviewed 2026-08-05 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Forecasting Low-Dimensional Turbulence via Multi-Dimensional Hybrid Quantum Reservoir Computing." pith.science (2026). https://pith.science/paper/HIG3WF4Z

@misc{pith2026250904006,
  author       = {Pith},
  title        = {Pith review of: Forecasting Low-Dimensional Turbulence via Multi-Dimensional Hybrid Quantum Reservoir Computing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HIG3WF4Z}},
  note         = {Machine review of arXiv:2509.04006}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

The prediction of complex dynamics remains an open problem across many domains of physics, where nonlinearities and multiscale interactions severely limit the reliability of conventional forecasting methods. Quantum reservoir computing (QRC) has emerged as a promising paradigm for information processing by exploiting the high dimensionality of the Hilbert space, where the dynamics of quantum systems take place. Here, we introduce a hybrid quantum-classical reservoir architecture capable of handling multivariate time series through quantum evolution combined with classical memory enhancement. Our model employs a five-qubit transverse-field Ising Hamiltonian with input-modulated dynamics and temporal multiplexing, enabling the encoding of input signals over multiple timescales. We apply this framework to two paradigmatic models of chaotic behavior in fluid dynamics, where multiscale dynamics and nonlinearities play a dominant role: a low-dimensional truncation of the two-dimensional Navier-Stokes equations and the Lorenz-63 system. By systematically scanning the quantum system's parameter space, we identify regions that maximize forecasting performance, as measured by the Valid Prediction Time. The observed robustness and reliable performances for both dynamical systems suggest that this hybrid quantum approach offers a flexible platform for modelling complex nonlinear time series.

Figures

Figures reproduced from arXiv: 2509.04006 by A. Giordano, A. Vinci, C. Gencarelli, C. Mastroianni, F. Carbone, F. D'Amore, F. Plastina, J. Settino, L. Mariani, L. Pontieri, L. Primavera, L. Salatino.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Bifurcation map obtained from kinetic energy [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Evolution of the phase space trajectories, projected [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

71 extracted references · 60 canonical work pages · 3 internal anchors

  1. [1]

    TURBIMECs: study of TURBulence In MEditerranean Cyclone events

    Overview of the analysis for the Lorenz-63 system To assess the versatility of the proposed hybrid- QRC model, we applied it to the Lorenz-63 system, a classical benchmark for deterministic chaos. The Lorenz system arises from a reduced model of Rayleigh–B´ enard convection and is defined by three coupled nonlinear differential equations (Eq. B1). For spe...

    work page 2022

  2. [2]

    echo state

    H. Jaeger, The “echo state” approach to analysing and training recurrent neural networks-with an erratum note, Bonn, Germany: German national research center for information technology gmd technical report 148, 13 (2001)

    work page 2001

  3. [4]

    Pathak, B

    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, Physical review letters 120, 024102 (2018)

    work page 2018

  4. [5]

    Maass, T

    W. Maass, T. Natschl¨ ager, and H. Markram, Real-time computing without stable states: A new framework for neural computation based on perturbations, Neural computation 14, 2531 (2002)

    work page 2002

  5. [6]

    Jaeger and H

    H. Jaeger and H. Haas, Harnessing nonlinearity: Predicting chaotic systems and saving energy in wireless communication, science 304, 78 (2004)

    work page 2004

  6. [7]

    Shor, Algorithms for quantum computation: discrete logarithms and factoring, in Proceedings 35th Annual Symposium on Foundations of Computer Science (1994) pp

    P. Shor, Algorithms for quantum computation: discrete logarithms and factoring, in Proceedings 35th Annual Symposium on Foundations of Computer Science (1994) pp. 124–134

    1994

  7. [8]

    L. K. Grover, A fast quantum mechanical algorithm for database search, in Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing , STOC ’96 (Association for Computing Machinery, New York, NY, USA, 1996) p. 212–219

    work page 1996

  8. [9]

    M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, 2010)

    2010

  9. [10]

    Montanaro, Quantum algorithms: an overview, npj Quantum Information 2, 1 (2016)

    A. Montanaro, Quantum algorithms: an overview, npj Quantum Information 2, 1 (2016)

    work page 2016

  10. [11]

    Preskill, Quantum computing in the nisq era and beyond, Quantum 2, 79 (2018)

    J. Preskill, Quantum computing in the nisq era and beyond, Quantum 2, 79 (2018)

    2018

  11. [12]

    Mastroianni, F

    C. Mastroianni, F. Plastina, L. Scarcello, J. Settino, and A. Vinci, Assessing quantum computing performance for energy optimization in a prosumer community, IEEE Transactions on Smart Grid 15, 444 (2023)

    work page 2023

  12. [13]

    Mastroianni, F

    C. Mastroianni, F. Plastina, J. Settino, and A. Vinci, Variational quantum algorithms for the allocation of resources in a cloud/edge architecture, IEEE Transactions on Quantum Engineering 5, 1 (2024)

    work page 2024

  13. [14]

    Consiglio, J

    M. Consiglio, J. Settino, A. Giordano, C. Mastroianni, F. Plastina, S. Lorenzo, S. Maniscalco, J. Goold, and T. J. Apollaro, Variational gibbs state preparation on noisy intermediate-scale quantum devices, Physical Review A 110, 012445 (2024)

    work page 2024

  14. [15]

    D’Amore, L

    F. D’Amore, L. Mariani, C. Mastroianni, F. Plastina, L. Salatino, J. Settino, and A. Vinci, Assessing projected quantum kernels for the classification of iot data, arXiv preprint arXiv:2505.14593 (2025)

    arXiv 2025

  15. [16]

    Fujii and K

    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 (Springer, 2021) pp. 423–450

    work page 2021

  16. [17]

    Ghosh, K

    S. Ghosh, K. Nakajima, T. Krisnanda, K. Fujii, and T. C. Liew, Quantum neuromorphic computing with reservoir computing networks, Advanced Quantum Technologies 4, 2100053 (2021)

    work page 2021

  17. [18]

    C. Zhu, P. J. Ehlers, H. I. Nurdin, and D. Soh, Practical few-atom quantum reservoir computing, Phys. Rev. Res. 7, 023290 (2025)

    work page 2025

  18. [19]

    Sannia, R

    A. Sannia, R. Mart ´ ınez-Pe˜ na, M. C. Soriano, G. L. Giorgi, and R. Zambrini, Dissipation as a resource for Quantum Reservoir Computing, Quantum 8, 1291 (2024)

    work page 2024

  19. [20]

    Mart ´ ınez-Pe˜ na, G

    R. Mart ´ ınez-Pe˜ na, G. L. Giorgi, J. Nokkala, M. C. Soriano, and R. Zambrini, Dynamical phase transitions in quantum reservoir computing, Physical Review Letters 127, 100502 (2021)

    work page 2021

  20. [21]

    Mart ´ ınez-Pe˜ na, J

    R. Mart ´ ınez-Pe˜ na, J. Nokkala, G. L. Giorgi, R. Zambrini, and M. C. Soriano, Information processing capacity of spin-based quantum reservoir computing systems, Cognitive Computation 15, 1440 (2023)

    work page 2023

  21. [22]

    G¨ otting, F

    N. G¨ otting, F. Lohof, and C. Gies, Exploring quantumness in quantum reservoir computing, Phys. Rev. A 108, 052427 (2023)

    work page 2023

  22. [24]

    Innocenti, S

    L. Innocenti, S. Lorenzo, I. Palmisano, A. Ferraro, M. Paternostro, and G. M. Palma, Potential and limitations of quantum extreme learning machines, Communications Physics 2023 6:1 6, 1 (2023)

    work page 2023

  23. [25]

    G. L. Monaco, M. Bertini, S. Lorenzo, and G. M. Palma, Quantum extreme learning of molecular potential energy surfaces and force fields, Machine Learning: Science and Technology 5, 035014 (2024)

    work page 2024

  24. [26]

    Vetrano, G

    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 Information 11, 20 (2025)

    work page 2025

  25. [27]

    Palacios, R

    A. Palacios, R. Mart ´ ınez-Pe˜ na, M. C. Soriano, G. L. Giorgi, and R. Zambrini, Role of coherence in many-body quantum reservoir computing, Communications Physics 7, 369 (2024)

    work page 2024

  26. [28]

    Kobayashi, Q

    S. Kobayashi, Q. H. Tran, and K. Nakajima, Extending echo state property for quantum reservoir computing, Phys. Rev. E 110, 024207 (2024)

    work page 2024

  27. [29]

    Monzani, E

    F. Monzani, E. Ricci, L. Nigro, and E. Prati, Leveraging non-unital noise for gate-based quantum reservoir computing, Arxiv (2024), arXiv:2409.07886v1

    Pith/arXiv arXiv 2024

  28. [30]

    M. N. Ivaki, A. Lazarides, and T. Ala-Nissila, Quantum reservoir computing on random regular graphs, ArXiv (2024), arXiv:2409.03665 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  29. [31]

    Fujii and K

    K. Fujii and K. Nakajima, Harnessing disordered- ensemble quantum dynamics for machine learning, Physical Review Applied 8, 024030 (2017)

    work page 2017

  30. [32]

    De Lorenzis, M

    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)

    work page 2025

  31. [33]

    Settino, G

    J. Settino, G. Luciano, A. Di Bartolomeo, P. Silvestrini, M. Lisitskiy, B. Ruggiero, and F. Romeo, Topology- enhanced superconducting qubit networks for in- sensor quantum information processing, arXiv preprint arXiv:2507.13228 (2025)

    work page arXiv 2025

  32. [34]

    Ghosh, A

    S. Ghosh, A. Opala, M. Matuszewski, T. Paterek, and T. C. H. Liew, Quantum reservoir processing, npj 9 Quantum Information 5, 35 (2019)

    work page 2019

  33. [35]

    Q. H. Tran, S. Ghosh, and K. Nakajima, Quantum- classical hybrid information processing via a single quantum system, Physical Review Research 5, 43127 (2023)

    work page 2023

  34. [36]

    Llodr` a, C

    G. Llodr` a, C. Charalambous, G. L. Giorgi, and R. Zambrini, Benchmarking the role of particle statistics in quantum reservoir computing, Advanced Quantum Technologies 6, 2200100 (2023)

    work page 2023

  35. [37]

    Romeo and J

    F. Romeo and J. Settino, Probing graph topology from local quantum measurements, arXiv preprint arXiv:2507.23689 (2025)

    work page arXiv 2025

  36. [38]

    L. C. Govia, G. J. Ribeill, G. E. Rowlands, H. K. Krovi, and T. A. Ohki, Quantum reservoir computing with a single nonlinear oscillator, Physical Review Research 3, 013077 (2021)

    work page 2021

  37. [39]

    Abbas and I

    A. Abbas and I. S. Maksymov, Reservoir computing using measurement-controlled quantum dynamics, Electronics 13, 1164 (2024)

    work page 2024

  38. [40]

    Suprano, D

    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)

    work page 2024

  39. [41]

    R. A. Bravo, K. Najafi, X. Gao, and S. F. Yelin, Quantum reservoir computing using arrays of rydberg atoms, PRX Quantum 3, 030325 (2022)

    work page 2022

  40. [42]

    Kornjaˇ ca, H.-Y

    M. Kornjaˇ ca, H.-Y. Hu, C. Zhao, J. Wurtz, P. Weinberg, M. Hamdan, A. Zhdanov, S. H. Cantu, H. Zhou, R. A. Bravo, et al. , Large-scale quantum reservoir learning with an analog quantum computer, arXiv preprint arXiv:2407.02553 (2024)

    Pith/arXiv arXiv 2024

  41. [43]

    Domingo, G

    L. Domingo, G. Carlo, and F. Borondo, Taking advantage of noise in quantum reservoir computing, Scientific Reports 2023 13:1 13, 1 (2023)

    work page 2023

  42. [44]

    Domingo, M

    L. Domingo, M. Grande, G. Carlo, F. Borondo, and J. Borondo, Optimal quantum reservoir computing for market forecasting: An application to fight food price crises, ArXiv (2023), arXiv:2401.03347 [quant-ph]

    Pith/arXiv arXiv 2023

  43. [45]

    Yasuda, Y

    T. Yasuda, Y. Suzuki, T. Kubota, K. Nakajima, Q. Gao, W. Zhang, S. Shimono, H. I. Nurdin, and N. Yamamoto, Quantum reservoir computing with repeated measurements on superconducting devices, ArXiv (2023), arXiv:2310.06706 [quant-ph]

    Pith/arXiv arXiv 2023

  44. [46]

    Wudarski, D

    F. Wudarski, D. O‘Connor, S. Geaney, A. A. Asanjan, M. Wilson, E. Strbac, P. A. Lott, and D. Venturelli, Hybrid quantum-classical reservoir computing for simulating chaotic systems, ArXiv (2024), arXiv:2311.14105 [quant-ph]

    Pith/arXiv arXiv 2024

  45. [47]

    Kubota, Y

    T. Kubota, Y. Suzuki, S. Kobayashi, Q. H. Tran, N. Yamamoto, and K. Nakajima, Temporal information processing induced by quantum noise, Physical Review Research 5, 023057 (2023)

    work page 2023

  46. [48]

    F. G. Fuchs, A. J. Stasik, S. Miao, O. T. Kulseng, and R. P. Bassa, Quantum reservoir computing using the stabilizer formalism for encoding classical data, ArXiv (2024), arXiv:2407.00445 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  47. [49]

    Govia, G

    L. Govia, G. Ribeill, G. Rowlands, H. Krovi, and T. Ohki, Quantum reservoir computing with a single nonlinear oscillator, Physical Review Research 3, 013077 (2021)

    work page 2021

  48. [50]

    Kutvonen, K

    A. Kutvonen, K. Fujii, and T. Sagawa, Optimizing a quantum reservoir computer for time series prediction, Scientific reports 10, 14687 (2020)

    work page 2020

  49. [51]

    Settino, L

    J. Settino, L. Salatino, L. Mariani, M. Channab, L. Bozzolo, S. Vallisa, P. Barill` a, A. Policicchio, N. L. Gullo, A. Giordano, et al. , Memory-augmented hybrid quantum reservoir computing, arXiv preprint arXiv:2409.09886 (2024)

    Pith/arXiv arXiv 2024

  50. [52]

    Minimal Quantum Reservoirs with Hamiltonian Encoding

    G. McCaul, J. S. T. Gongora, W. Otieno, S. Savelev, A. Zagoskin, and A. G. Balanov, Minimal quantum reservoirs with hamiltonian encoding, arXiv preprint arXiv:2505.22575 (2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  51. [53]

    Carbone, D

    F. Carbone, D. Telloni, G. Zank, and L. Sorriso- Valvo, Transition to turbulence in a five-mode galerkin truncation of two-dimensional magnetohydrodynamics, Phys. Rev. E 104, 025201 (2021)

    work page 2021

  52. [54]

    Boffetta and R

    G. Boffetta and R. E. Ecke, Two-dimensional turbulence, Annu. Rev. Fluid Mech. 44, 427 (2012)

    work page 2012

  53. [55]

    E. N. Lorenz, Deterministic Nonperiodic Flow, Journal of the Atmospheric Sciences 20, 130 (1963)

    work page 1963

  54. [56]

    Steinegger and C

    J. Steinegger and C. R¨ ath, Predicting three-dimensional chaotic systems with four qubit quantum systems, Scientific Reports 15, 6201 (2025)

    work page 2025

  55. [57]

    A. E. Hoerl and R. W. Kennard, Ridge regression: Biased estimation for nonorthogonal problems, Technometrics 12, 55 (1970)

    1970

  56. [58]

    Boffetta and R

    G. Boffetta and R. E. Ecke, Two-dimensional turbulence, Annual Review of Fluid Mechanics 44, 427 (2012)

    work page 2012

  57. [59]

    Boldrighini and V

    C. Boldrighini and V. Franceschini, A five-dimensional truncation of the plane incompressible navier-stokes equations, Communications in Mathematical Physics 64, 159 (1979)

    work page 1979

  58. [60]

    Franceschini, C

    V. Franceschini, C. Tebaldi, and F. Zironi, Fixed point limit behavior of n-mode truncated navier-stokes equations as n increases, Journal of Statistical Physics 35, 387 (1984)

    work page 1984

  59. [61]

    Carbone, Francesco, Telloni, Daniele, Zank, Gary, and Sorriso-Valvo, Luca, Chaotic advection and particle pairs diffusion in a low-dimensional truncation of two- dimensional magnetohydrodynamics, EPL 138, 53001 (2022)

    work page 2022

  60. [62]

    H. K. Moffatt, Magnetic field generation in electrically conducting fluids, Cambridge monographs on mechanics and applied mathematics (Cambridge University Press, 1983)

    work page 1983

  61. [63]

    A. C. Ting, W. H. Matthaeus, and D. Montgomery, Turbulent relaxation processes in magnetohydrodynamics, The Physics of Fluids 29, 3261 (1986), https://aip.scitation.org/doi/pdf/10.1063/1.865843

    work page doi:10.1063/1.865843 1986

  62. [64]

    R. D. Bartolo and V. Carbone, The role of the basic three-modes interaction during the free decay of magnetohydrodynamic turbulence, Europhysics Letters (EPL) 73, 547 (2006)

    work page 2006

  63. [65]

    Carbone and D

    F. Carbone and D. Dutykh, Route to chaos and resonant triads interaction in a truncated rotating nonlinear shallow–water model, PLOS ONE 19, 1 (2024)

    work page 2024

  64. [66]

    Boffetta and S

    G. Boffetta and S. Musacchio, Evidence for the double cascade scenario in two-dimensional turbulence, Phys. Rev. E 82, 016307 (2010)

    work page 2010

  65. [67]

    Dormand and P

    J. Dormand and P. Prince, A family of embedded runge- kutta formulae, Journal of Computational and Applied Mathematics 6, 19 (1980)

    work page 1980

  66. [68]

    Ernst, W

    H. Ernst, W. Gerhard, and P. N. Syvert, Solving Ordinary Differential Equations I: Nonstiff Problems , 2nd ed., Springer Series in Computational Mathematics 8 (Springer-Verlag Berlin Heidelberg, 1993). 10

    work page 1993

  67. [69]

    Franceschini and C

    V. Franceschini and C. Tebaldi, Sequences of infinite bifurcations and turbulence in a five-mode truncation of the navier-stokes equations, Journal of Statistical Physics 21, 707 (1979)

    work page 1979

  68. [70]

    M. J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, Journal of Statistical Physics 19, 25 (1978)

    work page 1978

  69. [71]

    Vlachas, J

    P. Vlachas, J. Pathak, B. Hunt, T. Sapsis, M. Girvan, E. Ott, and P. Koumoutsakos, Backpropagation algorithms and reservoir computing in recurrent neural networks for the forecasting of complex spatiotemporal dynamics, Neural Networks 126, 191 (2020)

    work page 2020

  70. [72]

    Franceschini, A feigenbaum sequence of bifurcations in the lorenz model, Journal of Statistical Physics 22, 397–406 (1980)

    V. Franceschini, A feigenbaum sequence of bifurcations in the lorenz model, Journal of Statistical Physics 22, 397–406 (1980)

    work page 1980

  71. [73]

    Ahmed, F

    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). 11 Appendix A: Navier-Stokes The analysis of the heatmap in Fig. 7 (Top) reveals the existence of preferential time scales for which the predictive performance is significantly enha...

    work page 2024

This paper was first reviewed by deepseek-v4-flash on August 5, 2026.