Pith. sign in

REVIEW 3 major objections 7 minor 51 references

Reviewed by Pith at T0; open to challenge.

T0 means a machine referee read the full paper against a public rubric. The mark states how deep the mechanical check went, never who wrote it. the ladder, T0–T4 →

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

T0 review · glm-5.2

GHZ-enriched quantum reservoir predicts chaotic PDEs an order of magnitude better

2026-07-08 03:34 UTC pith:TJDDZ3BQ

load-bearing objection GHZ-initialized quantum reservoir network shows order-of-magnitude RMSE improvement over prior QRN designs on KS prediction, but the causal link to metrological sensitivity is unproven and the headline result is a single best run. the 3 major comments →

arxiv 2607.06500 v1 pith:TJDDZ3BQ submitted 2026-07-07 quant-ph

Leveraging Metrologically Useful States in Quantum Reservoir Networks

Erik L. Connerty , Margarite LaBorde , Ethan N. Evans This is my paper
classification quant-ph
keywords quantum reservoir computingGHZ statequantum metrologyKuramoto-Sivashinsky equationquantum Fisher informationecho state networkpartial differential equation predictionautoencoder
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.

This paper presents a quantum reservoir network (QRN) augmented with a GHZ state preparation unitary — borrowed from quantum metrology — to predict the chaotic Kuramoto-Sivashinsky (KS) PDE. The authors compress the 128-dimensional KS system to 4 latent dimensions via a classical convolutional autoencoder, then feed those latent variables into a 16-qubit QRN that applies a GHZ state preparation circuit at each timestep. They report that the GHZ-enhanced QRN achieves an order-of-magnitude improvement in root mean square error over alternative QRN designs (sparse, no-GHZ, and random-state-preparation baselines) on the latent-space prediction task. They further show that the QRN outperforms classical echo-state networks when no weight regularization is used, and that it generalizes better over longer forecasting horizons. The authors motivate the GHZ unitary by arguing that its metrological sensitivity (Heisenberg-scaled precision) should translate into richer, more informative measurement distributions for the reservoir's feature vector, and they provide quantum Fisher information (QFI) calculations showing higher per-parameter QFI for the GHZ configuration in the pure-state case. The paper also documents a practical pitfall: the autoencoder appears to linearize the KS dynamics, which complicates model comparison in latent space.

Core claim

The central claim is that preparing a metrologically useful GHZ state at each timestep of a quantum reservoir network substantially improves the reservoir's performance on a chaotic time-series prediction task, and that this improvement is connected to the state's enhanced quantum Fisher information. The GHZ unitary is the key architectural addition: it biases the evolving quantum state toward a symmetric superposition that distributes amplitude more evenly across the computational basis, reducing sampling noise on rarely-visited basis states and, the authors argue, increasing the circuit's sensitivity to input-dependent parameter changes. The result is an order-of-magnitude RMSE reduction (

What carries the argument

GHZ state preparation unitary applied at each recurrent timestep of a 16-qubit quantum reservoir network, with weak measurement and reset on half the qubits preserving memory across timesteps; a convolutional autoencoder compresses 128-dimensional KS data to 4 latent dimensions before quantum processing.

Load-bearing premise

The paper assumes that the GHZ state's metrological sensitivity is causally responsible for the observed performance improvement, but this is not proven: the GHZ circuit also adds an extra unitary at each timestep, so the gain could stem from increased circuit depth or altered dynamics rather than metrological sensitivity per se, and the mixed-state QFI analysis meant to support the claim is described by the authors as inconclusive due to high variance.

What would settle it

Construct a QRN variant that matches the GHZ circuit's depth and gate count but uses a non-metrologically-useful entangled state (e.g., a generic entangled state with low QFI); if this variant matches the GHZ QRN's prediction accuracy, the metrological-sensitivity mechanism is not the operative cause.

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

If this is right

  • If metrologically useful states genuinely enhance quantum reservoir computing, the same principle could extend to other entangled states known to achieve sub-shot-noise sensitivity, potentially opening a design space for reservoir circuits guided by QFI maximization rather than heuristic architecture search.
  • The finding that the QRN generalizes better without regularization while classical ESNs require it suggests that quantum reservoirs may impose an implicit regularization through their physical dynamics, which could reduce hyperparameter tuning burdens in practice.
  • The autoencoder-induced linearization of KS dynamics flagged by the authors implies that benchmarking quantum vs. classical models on latent-space predictions may systematically obscure or inflate differences, motivating direct comparisons in the original PDE space when qubit counts permit.
  • The shot-scaling analysis suggests that GHZ-prepared reservoirs converge to stable output distributions faster, which if confirmed on hardware would reduce the measurement budget needed for near-term quantum ML applications.

Where Pith is reading between the lines

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

  • The causal link between metrological sensitivity and prediction performance is hypothesized but not isolated: the GHZ circuit also adds an extra unitary layer per timestep, so increased circuit depth or altered dynamics could independently explain the gains. A controlled ablation matching circuit depth without GHZ symmetry would test this.
  • The random-state-preparation baseline uses an efficient random circuit approximation rather than true Haar-random unitaries, so the comparison does not fully rule out the possibility that any sufficiently expressive state preparation would perform comparably.
  • If the autoencoder is linearizing the dynamics, the latent-space prediction task may be substantially easier than the original PDE prediction, meaning the QRN's advantage over classical methods could shrink or disappear when evaluated on the full 128-dimensional system without dimensionality reduction.

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

3 major / 7 minor

Summary. This paper presents a quantum reservoir network (QRN) augmented with a GHZ state preparation unitary for predicting latent-space representations of the 1-D Kuramoto-Sivashinsky (KS) chaotic PDE. The architecture combines a classical convolutional autoencoder (reducing 128 DOF to 4) with a 16-qubit QRN featuring input-dependent rotations, a reuploading block, and a GHZ unitary applied at each recurrent timestep. The authors report an order-of-magnitude RMSE improvement over a sparse QRN baseline and over no-GHZ and random-state-preparation variants, and show favorable comparison to classical echo-state networks (ESNs) in the unregularized setting. A QFI analysis is presented to connect the performance gain to the metrological properties of the GHZ state.

Significance. The empirical demonstration that GHZ state preparation consistently improves QRN performance across shot counts (Figure 7) is a useful contribution to the QRC literature, as is the transparent discussion of autoencoder-induced linearization of KS dynamics (Section 3.4). The shot-scaling comparison across four circuit variants is well-designed in principle. However, the paper's central causal claim—that metrological sensitivity is responsible for the improvement—is not substantiated by the evidence presented, and the headline RMSE comparison against classical baselines rests on a single QRN run versus averaged ESN runs. These issues significantly diminish the significance of the results as currently framed.

major comments (3)
  1. Section 2.1, GHZ Unitary: The paper's title and framing attribute the performance improvement to the metrological sensitivity of the GHZ state, but the evidence does not establish this causal link. The pure-state QFI result (Figure 8a) is nearly tautological—applying a GHZ unitary increases QFI by construction. The physically relevant test, mixed-state QFI after measurement and reset (Figure 8b, Eq. 7), is described as 'inconclusive' due to high variance. This is the test that would determine whether the metrological advantage survives the non-unitary reservoir dynamics, and it does not yield a usable result. The paper itself offers an alternative, non-metrological mechanism in Section 2.1: the GHZ unitary 'distributes amplitude more symmetrically across the computational basis,' producing a flatter output distribution that yields better-conditioned feature vectors. A simple control—a产品态
  2. Figure 9, caption: The QRN headline RMSE of 0.0162 is reported as a single best run ('computational complexity was prohibitively large and only allowed for the plotting of the best run'), while the classical ESN baselines are averaged over 10 seeds (Table 1, N_trials=10). This asymmetry undermines the quantitative comparison. Without variance estimates for the QRN, the reader cannot assess whether the QRN's advantage over the 8-node ESN is statistically meaningful or within run-to-run fluctuation. This is load-bearing for the claim of superior performance over classical methods.
  3. Section 3.2: The no-GHZ baseline removes the GHZ unitary but the GHZ variant also differs in that it applies an additional unitary at each timestep. The improvement could therefore stem from increased circuit depth or altered recurrent dynamics rather than from any metrological property of the GHZ state specifically. The random-state-prep comparison partially addresses this, but uses an 'efficient random circuit approximation' rather than true Haar-random unitaries, which the authors acknowledge. A control using a product-state superposition (e.g., Hadamard on each qubit) would isolate whether the benefit comes from entanglement/metrology or from distribution flattening, and is computationally cheap to implement.
minor comments (7)
  1. Section 2.1: 'We hypothesize that this symmetric state reduces the number of measurements needed to fully reconstruct our probability density function.' This hypothesis is never directly tested. The shot-scaling analysis in Figure 7 shows convergence behavior but does not isolate the measurement-efficiency claim.
  2. Section 3.4, Figure 10: The observation that linear regression outperforms both the QRN and ESNs at longer time horizons is striking and somewhat undercuts the motivation for using the QRN. The authors acknowledge this as a confound from autoencoder linearization, but it deserves more prominent discussion given that it affects the interpretation of all results in the paper.
  3. Equation (1): The notation for the weight tensor indices is slightly inconsistent with the surrounding text. The subscript structure W^{in}_{i,j,k,l} is introduced but the relationship between indices j,k,l and the qubit/gate/Euler-angle structure should be stated more explicitly.
  4. Figure 7: Error bars or variance bands are not shown for any circuit variant. Given that the random-state-prep results are averaged over multiple unitaries, some indication of spread would be informative.
  5. Section 3.3: The claim that 'the QRN generalizes better to the test data without any need for regularization' is strong given the single-run QRN data. Consider softening to reflect the uncertainty.
  6. Reference [3] is authored by Connerty (also an author on this manuscript). This prior-work relationship is disclosed via the citation but should be stated explicitly in the text for transparency.
  7. Typos: 'Reuploading' is misspelled as 're-euploading' in the Weight Initialization paragraph; 'genuine multipartite entanglement' in the Introduction is missing a space after 'of'.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The three major comments all identify legitimate weaknesses in the current framing of our results. We agree that (1) the causal link between metrological sensitivity and performance is not established by the evidence presented, (2) the QRN-vs-ESN comparison is asymmetric in its treatment of variance, and (3) a product-state control is needed to disentangle the contribution of entanglement/metrology from distribution flattening. We address each below and describe the revisions we will make.

read point-by-point responses

  1. Referee: Section 2.1, GHZ Unitary: The paper's title and framing attribute the performance improvement to the metrological sensitivity of the GHZ state, but the evidence does not establish this causal link. The pure-state QFI result (Figure 8a) is nearly tautological—applying a GHZ unitary increases QFI by construction. The physically relevant test, mixed-state QFI after measurement and reset (Figure 8b, Eq. 7), is described as 'inconclusive' due to high variance. The paper itself offers an alternative, non-metrological mechanism: the GHZ unitary distributes amplitude more symmetrically across the computational basis, producing a flatter output distribution. A product-state control would isolate the mechanism.

    Authors: The referee is correct that the current manuscript overstates the causal link between metrological sensitivity and the observed performance improvement. We concede the following: (a) the pure-state QFI result in Figure 8a is essentially confirmatory rather than evidentiary—applying a GHZ unitary increases QFI by construction, so this figure does not independently establish that metrological advantage is the operative mechanism in the reservoir setting. (b) The mixed-state QFI result (Figure 8b), which is the physically relevant test because it accounts for the non-unitary measurement-and-reset dynamics of the reservoir, is inconclusive due to high variance, as we acknowledged in the original text. (c) The manuscript itself offers an alternative, non-metrological mechanism in Section 2.1—namely, that the GHZ unitary distributes amplitude more symmetrically across the computational basis, yielding better-conditioned feature vectors—without testing this against a product-state control. We will revise the manuscript in two ways. First, we will soften the causal language throughout the paper (title, abstract, and main text) to accurately reflect what is and is not established. The title will be revised to remove the implication that metrological sensitivity is the demonstrated mechanism; a more accurate framing is that GHZ state preparation improves QRN performance and is motivated by metrological considerations, but the precise mechanism remains an open question. Second, we will add a product-state control (Hadamard on each qubit, producing a uniform product-state superposition) to the shot-scaling comparison in Section 3.2. This control is computationally cheap and directly tests whether the benefit comes from entanglement/metrology or from distribution flattening. If theH revision: no

  2. Referee: Figure 9, caption: The QRN headline RMSE of 0.0162 is reported as a single best run, while the classical ESN baselines are averaged over 10 seeds (Table 1, N_trials=10). This asymmetry undermines the quantitative comparison. Without variance estimates for the QRN, the reader cannot assess whether the QRN's advantage over the 8-node ESN is statistically meaningful or within run-to-run fluctuation.

    Authors: The referee is correct that the asymmetric treatment of variance between the QRN (single best run) and the ESN (averaged over 10 seeds) is a significant weakness in the quantitative comparison. We acknowledge that without variance estimates for the QRN, the reader cannot assess whether the advantage over the 8-node ESN is statistically meaningful. The computational cost of a single full QRN run at 960,000 shots over 5000 timesteps on a 16-qubit circuit was the limiting factor, but this does not excuse the absence of error bars or confidence intervals. We will address this in the revision by running the QRN with multiple independent random weight initializations (we estimate 5-10 seeds is feasible within our computational budget) and reporting the mean and standard deviation of the RMSE. We will update Figure 9 and its caption to present the QRN results with the same statistical treatment as the ESN baselines. If the computational budget does not permit 10 full seeds, we will report however many we can achieve and be transparent about the sample size. We will also add a note acknowledging this limitation explicitly in the text rather than burying it in the figure caption. revision: no

  3. Referee: Section 3.2: The no-GHZ baseline removes the GHZ unitary but the GHZ variant also differs in that it applies an additional unitary at each timestep. The improvement could stem from increased circuit depth or altered recurrent dynamics rather than from any metrological property of the GHZ state specifically. The random-state-prep comparison partially addresses this, but uses an 'efficient random circuit approximation' rather than true Haar-random unitaries. A control using a product-state superposition (e.g., Hadamard on each qubit) would isolate whether the benefit comes from entanglement/metrology or from distribution flattening, and is computationally cheap to implement.

    Authors: This is a well-taken point and is closely related to the first comment. We agree that the no-GHZ baseline does not fully control for the confound of additional circuit depth, since the GHZ variant applies an extra unitary at each timestep. The random-state-prep comparison was intended to address this by applying a comparable-depth unitary, but as the referee notes, it uses an efficient random circuit approximation rather than true Haar-random unitaries, which we acknowledged in the original text. The product-state superposition control (Hadamard on each qubit) that the referee suggests is the cleanest way to isolate whether the benefit comes from entanglement/metrology or from distribution flattening, and it is computationally cheap. We will implement this control and add it to the shot-scaling comparison in Section 3.2 (Figure 7) and to the decoded performance comparisons in Supplementary Note A. This will allow us to distinguish between three hypotheses: (1) the benefit comes from metrological sensitivity / entanglement (GHZ outperforms Hadamard product state), (2) the benefit comes from distribution flattening (Hadamard product state performs comparably to GHZ), or (3) the benefit comes from increased circuit depth alone (both GHZ and Hadamard outperform no-GHZ by similar margins). We will revise the discussion in Section 3.2 to present this three-way comparison and to draw conclusions accordingly. revision: no

Circularity Check

0 steps flagged

No significant circularity; one minor self-citation for baseline architecture and a near-tautological QFI diagnostic, but central claims are tested against independent baselines.

full rationale

The paper's central claim—that the GHZ-prepared QRN outperforms alternative implementations in RMSE on KS latent-space prediction—is tested against multiple baselines (no-GHZ QRN, Sparse QRN from [3], random-state-prep QRN, and classical ESNs) via direct simulation. The RMSE results are empirical measurements from simulation, not derived from fitted parameters or self-referential definitions. The self-citation to [3] (Connerty is an author on both) provides the baseline Sparse QRN architecture, but the improvement claim is grounded in the comparison runs, not in a result imported from [3]. The QFI analysis (Figure 8a) comes closest to being tautological: applying a GHZ state-preparation unitary and then computing high pure-state QFI is nearly guaranteed since GHZ states are known to have maximal QFI by construction. However, the paper presents this as a diagnostic observation rather than as a derivation or prediction, and it is honest that the physically relevant mixed-state QFI (Figure 8b) is 'inconclusive' due to high variance. The causal claim that metrological sensitivity is responsible for the RMSE improvement is under-supported (the skeptic correctly identifies distribution-flattening as an alternative mechanism), but this is a correctness/interpretation risk, not a circularity in the derivation chain. No step was found where a prediction reduces to its inputs by construction.

Axiom & Free-Parameter Ledger

5 free parameters · 4 axioms · 1 invented entities

The paper introduces no new physical entities or particles. The 'metrologically-useful state preparation' is an application of known GHZ states to a new context. The free parameters are primarily random initialization weights (standard in reservoir computing) and the autoencoder's 1M+ trained parameters. The key ad-hoc axiom is that metrological sensitivity transfers to ML feature quality, which is the paper's central hypothesis.

free parameters (5)
  • W_in = sampled from U(-π/C, π/C), C=dn_c
    Input weight tensor for rotation gates; randomly initialized but fixed during training. Not fitted to data in the gradient-descent sense, but sampled from a distribution with a chosen scaling constant.
  • W_bias = sampled from U(-π/(2n_c), π/(2n_c))
    Bias weight tensor for rotation gates; randomly initialized.
  • W_hidden, W_in_hidden, W_entangle, W_in_entangle = various uniform distributions
    Additional weight tensors for two-qubit gates; randomly initialized with chosen scaling.
  • Autoencoder parameters = 1,085,925 parameters
    The convolutional autoencoder has over 1M parameters trained on 9,250 KS data points. These are fitted to data and directly affect the latent space the QRN predicts.
  • Ridge regression alpha (ESN) = 0.1
    Regularization parameter chosen for classical ESN comparison; the QRN uses alpha=0.
axioms (4)
  • standard math GHZ states provide Heisenberg-limited sensitivity in quantum metrology
    Standard result from quantum metrology [35, 36, 37, 38]. Invoked in Section 2.1 to motivate GHZ state preparation in the QRN.
  • ad hoc to paper Similar metrological gains transfer from parameter estimation to machine learning feature extraction
    Section 2.1: 'As quantum machine learning also involves extracting information from input data, it is natural to expect that similar gains can be observed in this context.' This is the core hypothesis; it is not proven and is the weakest link in the argument.
  • domain assumption The autoencoder faithfully preserves the dynamical structure of the KS system
    The 128-to-4 dimensionality reduction is assumed to preserve enough dynamics for meaningful prediction. Section 3.4 reveals this is questionable: the autoencoder appears to linearize the dynamics.
  • domain assumption Efficient random circuit approximation is a sufficient proxy for Haar-random unitaries
    Section 3.2: true Haar-random unitaries are computationally expensive, so an efficient random circuit approximation [45] is used. The quality of this approximation affects the validity of the random-state-prep control.
invented entities (1)
  • Metrologically-useful state preparation in QRN context independent evidence
    purpose: To improve QRN performance by leveraging entanglement and Heisenberg-scaled sensitivity from quantum metrology
    The paper provides empirical evidence (Figure 7: order-of-magnitude RMSE improvement) and QFI analysis (Figure 8a: increased QFI for pure states). However, the causal mechanism (metrological sensitivity → ML performance) is not proven; the QFI mixed-state result is inconclusive.

pith-pipeline@v1.1.0-glm · 19194 in / 2944 out tokens · 483791 ms · 2026-07-08T03:34:49.259257+00:00 · methodology

0 comments
read the original abstract

Interest in using quantum computers for the purpose of predicting chaotic partial differential equations (PDEs) has been growing with the advent of newer low-error quantum computers and robust simulation tools. In this paper, we present a method that utilizes a quantum reservoir network (QRN) to predict latent space representations of the high-dimensional chaotic 1-D Kuramoto-Sivashinksy (KS) system. This hybrid approach takes advantage of advancements in classical machine learning (ML) through the use of a classical autoencoder as well as techniques from quantum metrology through the use of a unitary that creates metrologically-useful states. Through rigorous simulation and analysis, we show that the proposed method outperforms alternative QRN implementations without this metrologically-useful state preparation, and also show better performance than classical echo-state networks when weight regularization is not used. Finally, we bring to light potential issues that can arise when using autoencoders within QRC pipelines.

Figures

Figures reproduced from arXiv: 2607.06500 by Erik L. Connerty, Ethan N. Evans, Margarite LaBorde.

Figure 1
Figure 1. Figure 1: The GHZ QRN Circuit. As before, the inner block named the “Reuploading Block” is repeated [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: State creation circuit for the GHZ state, which is [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: 1-D Kuramoto Sivashinsky data. The top data is the original data with [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Latent space test data performance of the QRN with [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Decoded latent space representation of the KS-1D system compared to the ground truth decoded latents for [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: An analysis with the 4 different versions of the QRN [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 10
Figure 10. Figure 10: Prediction using varying time-horizons from [PITH_FULL_IMAGE:figures/full_fig_p007_10.png] view at source ↗
Figure 9
Figure 9. Figure 9: A comparison showing the differing performance [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: Plots showing both the expectation values over time and the probability densities over time of the [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: 11 [PITH_FULL_IMAGE:figures/full_fig_p011_3_1.png] view at source ↗
Figure 12
Figure 12. Figure 12: Plots depicting the decoded PDE prediction performance for the Sparse QRN. [PITH_FULL_IMAGE:figures/full_fig_p012_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Plots depicting the decoded PDE prediction performance for the QRN without GHZ State Prep. [PITH_FULL_IMAGE:figures/full_fig_p013_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Plots depicting the decoded PDE prediction performance for the QRN with Haar Random State Prep. [PITH_FULL_IMAGE:figures/full_fig_p014_14.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

51 extracted references · 51 canonical work pages · 6 internal anchors

  1. [1]

    Peter W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum com- puter.SIAM Journal on Computing, 26(5):1484–1509, October 1997. ISSN 1095-7111. doi: 10.1137/ s0097539795293172. URL http://dx.doi.org/10. 1137/S0097539795293172

    work page 1997

  2. [2]

    Lov K. Grover. A fast quantum mechanical algorithm for database search. InProceedings of the Twenty-Eighth An- nual ACM Symposium on Theory of Computing, STOC ’96, page 212–219, New York, NY , USA, 1996. Associa- tion for Computing Machinery. ISBN 0897917855. doi: 10.1145/237814.237866. URL https://doi.org/ 10.1145/237814.237866

    work page doi:10.1145/237814.237866 1996

  3. [3]

    Connerty, Ethan N

    Erik L. Connerty, Ethan N. Evans, Gerasimos Ange- latos, and Vignesh Narayanan. Predicting chaotic dy- namics on nisq hardware with quantum reservoir net- works.Communications Physics, May 2026. ISSN 2399-

    work page 2026

  4. [4]

    URL https: //doi.org/10.1038/s42005-026-02652-1

    doi: 10.1038/s42005-026-02652-1. URL https: //doi.org/10.1038/s42005-026-02652-1

    work page doi:10.1038/s42005-026-02652-1

  5. [5]

    ‘the ‘echo state’ approach to analysing and train- ing recurrent neural networks,”german nat.Res

    H Jaeger. ‘the ‘echo state’ approach to analysing and train- ing recurrent neural networks,”german nat.Res. Center Inf. Technol., GMD Rep, 148:43, 2001

    work page 2001

  6. [6]

    Experimental demonstra- tion of enhanced quantum tomography via quantum reser- voir processing.Quantum Science and Technology, 10 (3):035041, June 2025

    Tanjung Krisnanda, Pengtao Song, Adrian Copetudo, Clara Yun Fontaine, Tomasz Paterek, Timothy C H Liew, and Yvonne Y Gao. Experimental demonstra- tion of enhanced quantum tomography via quantum reser- voir processing.Quantum Science and Technology, 10 (3):035041, June 2025. ISSN 2058-9565. doi: 10. 1088/2058-9565/addffe. URL https://doi.org/10. 1088/2058-...

    work page 2025

  7. [7]

    Large-scale quantum reservoir learning with an analog quantum computer

    Milan Kornjaˇca, Hong-Ye Hu, Chen Zhao, Jonathan Wurtz, Phillip Weinberg, Majd Hamdan, Andrii Zhdanov, Ser- gio H Cantu, Hengyun Zhou, Rodrigo Araiza Bravo, et al. Large-scale quantum reservoir learning with an analog quantum computer.arXiv:2407.02553, 2024

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  8. [8]

    Khan, Nicholas T

    Fangjun Hu, Saeed A. Khan, Nicholas T. Bronn, Gerasimos Angelatos, Graham E. Rowlands, Guilhem J. Ribeill, and Hakan E. Türeci. Overcoming the coherence time barrier in quantum machine learning on temporal data.Nature Communications, 15(1):7491, Aug 2024. ISSN 2041-1723. doi: 10.1038/s41467-024-51162-7. URL https://doi. org/10.1038/s41467-024-51162-7

    work page doi:10.1038/s41467-024-51162-7 2024

  9. [9]

    Osama Ahmed, Felix Tennie, and Luca Magri. Robust quantum reservoir computers for forecasting chaotic dy- namics: generalized synchronization and stability.Pro- ceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 481(2324):20250550, 10 2025. ISSN 1364-5021. doi: 10.1098/rspa.2025.0550. URL https://doi.org/10.1098/rspa.2025.0550

    work page doi:10.1098/rspa.2025.0550 2025

  10. [10]

    Feedback connections in quan- tum reservoir computing with mid-circuit measurements

    Jakob Murauer, Rajiv Krishnakumar, Sabine Tornow, and Michaela Geierhos. Feedback connections in quan- tum reservoir computing with mid-circuit measurements. In2025 IEEE International Conference on Quantum Computing and Engineering (QCE), page 1646–1652. IEEE, August 2025. doi: 10.1109/qce65121.2025.00182. URL http://dx.doi.org/10.1109/QCE65121. 2025.00182

    work page doi:10.1109/qce65121.2025.00182 2025

  11. [11]

    Feedback-driven quantum reservoir com- puting for time-series analysis.PRX Quantum, 5: 040325, Nov 2024

    Kaito Kobayashi, Keisuke Fujii, and Naoki Ya- mamoto. Feedback-driven quantum reservoir com- puting for time-series analysis.PRX Quantum, 5: 040325, Nov 2024. doi: 10.1103/PRXQuantum.5. 040325. URL https://link.aps.org/doi/10. 1103/PRXQuantum.5.040325

    work page doi:10.1103/prxquantum.5 2024

  12. [12]

    Enhancing the performance of quantum reser- voir computing and solving the time-complexity prob- lem by artificial memory restriction.Phys

    Saud ˇCindrak, Brecht Donvil, Kathy Lüdge, and Lina Jaurigue. Enhancing the performance of quantum reser- voir computing and solving the time-complexity prob- lem by artificial memory restriction.Phys. Rev. Res., 6:013051, Jan 2024. doi: 10.1103/PhysRevResearch.6. 013051. URL https://link.aps.org/doi/10. 1103/PhysRevResearch.6.013051

    work page doi:10.1103/physrevresearch.6 2024

  13. [13]

    Packet switching in quantum networks: A path to the quantum internet,

    Philipp Pfeffer, Florian Heyder, and Jörg Schu- macher. Hybrid quantum-classical reservoir comput- ing of thermal convection flow.Phys. Rev. Res., 4: 033176, Sep 2022. doi: 10.1103/PhysRevResearch.4. 033176. URL https://link.aps.org/doi/10. 1103/PhysRevResearch.4.033176

    work page doi:10.1103/physrevresearch.4 2022

  14. [14]

    Soriano, and Roberta Zambrini

    Pere Mujal, Rodrigo Martínez-Peña, Gian Luca Giorgi, Miguel C. Soriano, and Roberta Zambrini. Time- series quantum reservoir computing with weak and pro- jective measurements.npj Quantum Information, 9 8 (1):16, Feb 2023. ISSN 2056-6387. doi: 10.1038/ s41534-023-00682-z. URL https://doi.org/10. 1038/s41534-023-00682-z

    work page 2023

  15. [15]

    Quantum reservoir computing with repeated measurements on superconducting devices

    Toshiki Yasuda, Yudai Suzuki, Tomoyuki Kubota, Kohei Nakajima, Qi Gao, Wenlong Zhang, Satoshi Shimono, Hen- dra I. Nurdin, and Naoki Yamamoto. Quantum reservoir computing with repeated measurements on superconduct- ing devices, 2023. URL https://arxiv.org/abs/ 2310.06706

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  16. [16]

    Feedback-enhanced quantum reservoir computing with weak measurements

    Tomoya Monomi, Wataru Setoyama, and Yoshihiko Hasegawa. Feedback-enhanced quantum reservoir com- puting with weak measurements, 2025. URL https: //arxiv.org/abs/2503.17939

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  17. [17]

    Ehlers, Hendra I

    Chuanzhou Zhu, Peter J. Ehlers, Hendra I. Nurdin, and Daniel Soh. Minimalistic and scalable quantum reser- voir computing enhanced with feedback.npj Quantum Information, 11(1):195, Nov 2025. ISSN 2056-6387. doi: 10.1038/s41534-025-01144-4. URL https://doi. org/10.1038/s41534-025-01144-4

    work page doi:10.1038/s41534-025-01144-4 2025

  18. [18]

    Experimen- tal memory control in continuous-variable optical quan- tum reservoir computing.Nat

    Iris Paparelle, Johan Henaff, Jorge García-Beni, Émilie Gillet, Daniel Montesinos, Gian Luca Giorgi, Miguel C So- riano, Roberta Zambrini, and Valentina Parigi. Experimen- tal memory control in continuous-variable optical quan- tum reservoir computing.Nat. Photonics, 20(4):413–420, March 2026

    work page 2026

  19. [19]

    Performance study of variational quantum algorithms for solving the poisson equation on a quantum computer.Phys

    Mazen Ali and Matthias Kabel. Performance study of variational quantum algorithms for solving the poisson equation on a quantum computer.Phys. Rev. Appl., 20: 014054, Jul 2023. doi: 10.1103/PhysRevApplied.20. 014054. URL https://link.aps.org/doi/10. 1103/PhysRevApplied.20.014054

    work page doi:10.1103/physrevapplied.20 2023

  20. [20]

    N. M. Guseynov, A. A. Zhukov, W. V . Pogosov, and A. V . Lebedev. Depth analysis of variational quan- tum algorithms for the heat equation.Phys. Rev. A, 107:052422, May 2023. doi: 10.1103/PhysRevA.107. 052422. URL https://link.aps.org/doi/10. 1103/PhysRevA.107.052422

    work page doi:10.1103/physreva.107 2023

  21. [21]

    Y .Y . Liu, Z. Chen, C. Shu, P. Rebentrost, Y .G. Liu, S.C. Chew, B.C. Khoo, and Y .D. Cui. A varia- tional quantum algorithm-based numerical method for solving potential and stokes flows.Ocean Engi- neering, 292:116494, 2024. ISSN 0029-8018. doi: https://doi.org/10.1016/j.oceaneng.2023.116494. URL https://www.sciencedirect.com/science/ article/pii/S00298...

    work page doi:10.1016/j.oceaneng.2023.116494 2024

  22. [22]

    Watts, Mudassir Moosa, Yilian Liu, and Peter L

    Abhijat Sarma, Thomas W. Watts, Mudassir Moosa, Yilian Liu, and Peter L. McMahon. Quantum variational solving of nonlinear and multidimensional partial differential equa- tions.Phys. Rev. A, 109:062616, Jun 2024. doi: 10.1103/ PhysRevA.109.062616. URL https://link.aps. org/doi/10.1103/PhysRevA.109.062616

    work page doi:10.1103/physreva.109.062616 2024

  23. [23]

    Available: https://link.aps.org/doi/10.1103/PhysRevA

    Josephine Hunout, Sylvain Laizet, and Lorenzo Iannucci. Variational quantum algorithm based on lagrange poly- nomial encoding to solve differential equations.Phys. Rev. A, 111:062404, Jun 2025. doi: 10.1103/PhysRevA. 111.062404. URL https://link.aps.org/doi/ 10.1103/PhysRevA.111.062404

    work page doi:10.1103/physreva 2025

  24. [24]

    A variational quantum algo- rithm for tackling multi-dimensional poisson equations with inhomogeneous boundary conditions.New Jour- nal of Physics, 27(5):054510, may 2025

    Minjin Choi and Hoon Ryu. A variational quantum algo- rithm for tackling multi-dimensional poisson equations with inhomogeneous boundary conditions.New Jour- nal of Physics, 27(5):054510, may 2025. doi: 10.1088/ 1367-2630/add8b4. URL https://dx.doi.org/10. 1088/1367-2630/add8b4

    work page 2025

  25. [25]

    Varia- tional quantum algorithms for poisson equations based on the decomposition of sparse hamiltonians.Phys

    Hui-Min Li, Zhi-Xi Wang, and Shao-Ming Fei. Varia- tional quantum algorithms for poisson equations based on the decomposition of sparse hamiltonians.Phys. Rev. A, 108:032418, Sep 2023. doi: 10.1103/PhysRevA.108. 032418. URL https://link.aps.org/doi/10. 1103/PhysRevA.108.032418

    work page doi:10.1103/physreva.108 2023

  26. [26]

    McDermott, and Juan José Mendoza-Arenas

    Hirad Alipanah, Feng Zhang, Yongxin Yao, Richard Thompson, Nam Nguyen, Junyu Liu, Peyman Givi, Brian J. McDermott, and Juan José Mendoza-Arenas. Quan- tum dynamics simulation of the advection-diffusion equa- tion, 2025. URL https://arxiv.org/abs/2503. 13729

    work page 2025

  27. [27]

    Bryngelson

    Zhixin Song, Robert Deaton, Bryan Gard, and Spencer H. Bryngelson. Incompressible navier–stokes solve on noisy quantum hardware via a hybrid quantum–classical scheme. Computers & Fluids, 288:106507, 2025. ISSN 0045-7930. doi: https://doi.org/10.1016/j.compfluid.2024.106507. URL https://www.sciencedirect.com/ science/article/pii/S0045793024003384

    work page doi:10.1016/j.compfluid.2024.106507 2025

  28. [28]

    Yuan Chen, Abdul Khaliq, and Khaled M. Furati. Quan- tum recurrent neural networks with encoder-decoder for time-dependent partial differential equations, 2025. URL https://arxiv.org/abs/2502.13370

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  29. [29]

    Makoto Takagi, Ryuji Kokubo, Misato Kurosawa, Tsubasa Ikami, Yasuhiro Egami, Hiroki Nagai, Takahiro Kashikawa, Koichi Kimura, Yutaka Takita, and Yu Matsuda. Time- series forecasting for nonlinear high-dimensional system using hybrid method combining autoencoder and multi- parallelized quantum long short-term memory and gated re- current unit, 2025. URL ht...

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  30. [30]

    Diffusion-induced chaos in reaction systems.Progress of Theoretical Physics Supplement, 64: 346–367, 02 1978

    Yoshiki Kuramoto. Diffusion-induced chaos in reaction systems.Progress of Theoretical Physics Supplement, 64: 346–367, 02 1978. ISSN 0375-9687. doi: 10.1143/PTPS. 64.346. URL https://doi.org/10.1143/PTPS. 64.346

    work page doi:10.1143/ptps 1978

  31. [31]

    Sivashinsky

    G.I. Sivashinsky. Nonlinear analysis of hydrodynamic in- stability in laminar flames—I. derivation of basic equations. Acta Astronautica, 4(11):1177–1206, 1977. ISSN 0094-

    work page 1977

  32. [32]

    URL https://www.sciencedirect.com/ science/article/pii/0094576577900960

    doi: https://doi.org/10.1016/0094-5765(77)90096-0. URL https://www.sciencedirect.com/ science/article/pii/0094576577900960. 9

    work page doi:10.1016/0094-5765(77)90096-0

  33. [33]

    Multipartite entanglement and high-precision metrology.Phys

    Géza Tóth. Multipartite entanglement and high-precision metrology.Phys. Rev. A, 85:022322, Feb 2012. doi: 10.1103/PhysRevA.85.022322. URL https://link. aps.org/doi/10.1103/PhysRevA.85.022322

    work page doi:10.1103/physreva.85.022322 2012

  34. [34]

    Activating hidden metrological usefulness

    Géza Tóth, Tamás Vértesi, Paweł Horodecki, and Ryszard Horodecki. Activating hidden metrological usefulness. Phys. Rev. Lett., 125:020402, Jul 2020. doi: 10.1103/ PhysRevLett.125.020402. URL https://link.aps. org/doi/10.1103/PhysRevLett.125.020402

    work page doi:10.1103/physrevlett.125.020402 2020

  35. [35]

    Buchanan, and Amir Barati Farimani

    Zijie Li, Saurabh Patil, Francis Ogoke, Dule Shu, Wilson Zhen, Michael Schneier, John R. Buchanan, and Amir Barati Farimani. Latent neural PDE solver: A reduced-order modeling framework for partial differential equations.Journal of Computa- tional Physics, 524:113705, 2025. ISSN 0021-9991. doi: https://doi.org/10.1016/j.jcp.2024.113705. URL https://www.sc...

    work page doi:10.1016/j.jcp.2024.113705 2025

  36. [36]

    Katiana Kontolati, Somdatta Goswami, George Em Kar- niadakis, and Michael D. Shields. Learning nonlinear operators in latent spaces for real-time predictions of com- plex dynamics in physical systems.Nature Communi- cations, 15(1):5101, Jun 2024. ISSN 2041-1723. doi: 10.1038/s41467-024-49411-w. URL https://doi. org/10.1038/s41467-024-49411-w

    work page doi:10.1038/s41467-024-49411-w 2024

  37. [37]

    Leibfried, M

    D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland. Toward Heisenberg-limited spec- troscopy with multiparticle entangled states.Science, 304 (5676):1476–1478, 2004. doi: 10.1126/science.1097576. URL https://www.science.org/doi/abs/10. 1126/science.1097576

    work page doi:10.1126/science.1097576 2004

  38. [38]

    Quantum metrology from a quantum information science perspective.Jour- nal of Physics A: Mathematical and Theoretical, 47 (42):424006, oct 2014

    Géza Tóth and Iagoba Apellaniz. Quantum metrology from a quantum information science perspective.Jour- nal of Physics A: Mathematical and Theoretical, 47 (42):424006, oct 2014. doi: 10.1088/1751-8113/47/42/ 424006. URL https://dx.doi.org/10.1088/ 1751-8113/47/42/424006

    work page doi:10.1088/1751-8113/47/42/ 2014

  39. [39]

    Keaveney, A

    Luca Pezzé and Augusto Smerzi. Entanglement, non- linear dynamics, and the Heisenberg limit.Phys. Rev. Lett., 102:100401, Mar 2009. doi: 10.1103/PhysRevLett. 102.100401. URL https://link.aps.org/doi/ 10.1103/PhysRevLett.102.100401

    work page doi:10.1103/physrevlett 2009

  40. [40]

    Quantum metrology.Phys

    Vittorio Giovannetti, Seth Lloyd, and Lorenzo Mac- cone. Quantum metrology.Phys. Rev. Lett., 96: 010401, Jan 2006. doi: 10.1103/PhysRevLett.96. 010401. URL https://link.aps.org/doi/10. 1103/PhysRevLett.96.010401

    work page doi:10.1103/physrevlett.96 2006

  41. [41]

    Not all pure entangled states are useful for sub-shot-noise in- terferometry.Phys

    Philipp Hyllus, Otfried Gühne, and Augusto Smerzi. Not all pure entangled states are useful for sub-shot-noise in- terferometry.Phys. Rev. A, 82:012337, Jul 2010. doi: 10.1103/PhysRevA.82.012337. URL https://link. aps.org/doi/10.1103/PhysRevA.82.012337

    work page doi:10.1103/physreva.82.012337 2010

  42. [42]

    Deep residual learning for image recognition

    Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 770–778, 2016. doi: 10.1109/ CVPR.2016.90

    work page 2016

  43. [43]

    Kuramoto-sivashinsky solver in JAX based on exponential time differencing (etd)

    Felix Köhler. Kuramoto-sivashinsky solver in JAX based on exponential time differencing (etd). URL https://github.com/Ceyron/ machine-learning-and-simulation/blob/ main/english/fft_and_spectral_methods/ ks_solver_etd_in_jax.ipynb

    work page

  44. [44]

    Efficiency of produc- ing random unitary matrices with quantum circuits.Phys

    Ludovic Arnaud and Daniel Braun. Efficiency of produc- ing random unitary matrices with quantum circuits.Phys. Rev. A, 78:062329, Dec 2008. doi: 10.1103/PhysRevA.78. 062329. URL https://link.aps.org/doi/10. 1103/PhysRevA.78.062329

    work page doi:10.1103/physreva.78 2008

  45. [45]

    Random quantum circuits are approx- imate unitary t-designs in depth O(nt 5+o(1)).Quan- tum, 6:795, September 2022

    Jonas Haferkamp. Random quantum circuits are approx- imate unitary t-designs in depth O(nt 5+o(1)).Quan- tum, 6:795, September 2022. ISSN 2521-327X. doi: 10. 22331/q-2022-09-08-795. URL https://doi.org/ 10.22331/q-2022-09-08-795

    work page doi:10.22331/q-2022-09-08-795 2022

  46. [46]

    Fernando G. S. L. Brandão, Aram W. Harrow, and Michał Horodecki. Local random quantum circuits are approx- imate polynomial-designs.Communications in Mathe- matical Physics, 346(2):397–434, Sep 2016. ISSN 1432-

    work page 2016

  47. [47]

    URL https: //doi.org/10.1007/s00220-016-2706-8

    doi: 10.1007/s00220-016-2706-8. URL https: //doi.org/10.1007/s00220-016-2706-8

    work page doi:10.1007/s00220-016-2706-8

  48. [48]

    Random unitaries in extremely low depth.Science, 389(6755):92–96, 2025

    Thomas Schuster, Jonas Haferkamp, and Hsin-Yuan Huang. Random unitaries in extremely low depth.Science, 389(6755):92–96, 2025. doi: 10.1126/science.adv8590. URL https://www.science.org/doi/abs/10. 1126/science.adv8590

    work page doi:10.1126/science.adv8590 2025

  49. [49]

    Hoerl and Robert W

    Arthur E. Hoerl and Robert W. Kennard. Ridge regres- sion: Biased estimation for nonorthogonal problems.Tech- nometrics, 42(1):80–86, 2000. ISSN 00401706. URL http://www.jstor.org/stable/1271436

    work page arXiv 2000

  50. [50]

    Nathan Kutz, and Steven L

    Bethany Lusch, J. Nathan Kutz, and Steven L. Brun- ton. Deep learning for universal linear embeddings of nonlinear dynamics.Nature Communications, 9(1): 4950, Nov 2018. ISSN 2041-1723. doi: 10.1038/ s41467-018-07210-0. URL https://doi.org/10. 1038/s41467-018-07210-0

    work page 2018

  51. [51]

    Forecasting Sequential Data using Consistent Koopman Autoencoders

    Omri Azencot, N. Benjamin Erichson, Vanessa Lin, and Michael W. Mahoney. Forecasting sequential data using consistent koopman autoencoders, 2020. URL https: //arxiv.org/abs/2003.02236. 10 A Supplementary Note: Observations & Additional Results A.1 Sampled Observations Data from the QRN circuit is gathered as both probability distributions and expectation ...

    work page internal anchor Pith review Pith/arXiv arXiv 2020