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REVIEW 4 major objections 5 minor 27 references

A matrix product state classifier reveals an interpretable hierarchy of wildfire susceptibility classes, with non-adjacent categories more separable than adjacent ones—measured through quantum-inspired reduced-density-matrix diagnostics.

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2026-08-01 12:32 UTC pith:N7D7Y22D

load-bearing objection The mixedness math is sound, but the headline claim is confounded by ordinal bins of a modeled product and the lack of a same-features classical baseline. the 4 major comments →

arxiv 2607.19503 v1 pith:N7D7Y22D submitted 2026-07-21 physics.soc-ph cs.LGphysics.data-an

Tensor Network Machine Learning for Wildfire Susceptibility Mapping: from Grokking Dynamics to Quantum Mixedness of Class Representations

Domenico Pomarico , Alessandra Costantino , Gabriel Ramirez Sanchez , Loredana Bellantuono , Davide D' Al\`o , Mario Elia , Alessandro Fania , Francesco Giordano
show 7 more authors
Niloofar Kheirkhahan Raffaele Lafortezza Ester Pantaleo Sabina Tangaro Roberto Bellotti Alfonso Monaco Nicola Amoroso
This is my paper
classification physics.soc-ph cs.LGphysics.data-an
keywords tensor networksmatrix product stateswildfire susceptibilitygrokkingreduced density matricesclass separabilitygeospatial embeddingsquantum-inspired machine learning
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 tries to establish that tensor network models, specifically matrix product state (MPS) classifiers, are not just competitive predictors for wildfire susceptibility but also provide a physically grounded way to inspect how well classes are separated. Training on satellite-derived embeddings of the Gargano region, the MPS shows a delayed abrupt generalization gain (grokking) in the binary task, and in the four-class task it produces class-representation density matrices whose mixedness diagnostics reveal a consistent hierarchy: non-adjacent susceptibility levels are more separable than adjacent ones. If right, this gives a new interpretability tool—level-resolved mixedness—that quantifies inter-class confusion in environmental classification without relying on conventional feature-importance lists.

Core claim

The central discovery claimed is that a trained MPS classifier encodes class distinguishability in the off-diagonal structure of its label-space reduced density matrix. The authors show that pairwise decoherence factors, normalized triplet determinants, and conditional purities all converge to a stable ordering in which adjacent susceptibility classes (e.g., 1–2, 2–3) retain higher coherence and overlap, while non-adjacent pairs (e.g., 0–2, 0–3) are more decohered and therefore more distinguishable. They also report that the binary 0-vs-rest separation reproduces almost exactly the class-0 magnetization pattern in the multiclass setting, indicating a hierarchical, compositional representatio

What carries the argument

Matrix Product State (MPS) classifier—a tensor-train network that encodes each input as a product of local qubit states and contracts a class-dependent variational tensor (the 'quantum mask') to produce prediction amplitudes. The relevant diagnostics are built from the reduced density matrix in label space: pairwise decoherence factors Γij, normalized triplet determinants Δijk (squared volumes of conditional-state parallelepipeds), and conditional purities. These quantities track how much information about each class pair or triplet leaks into the environment, providing the claimed hierarchy of distinguishability.

Load-bearing premise

The four-class wildfire susceptibility labels are treated as ground truth; they are generated by multiplying two predicted maps and reclassifying into ordered bins, so if adjacent bins are inherently more similar, the claimed adjacency separability hierarchy could be an artifact of that label design.

What would settle it

Take the same learned geospatial embeddings and the same MPS training procedure, but shuffle the class labels (or relabel the four bins in a random order) before training; if the mixedness diagnostics then track the permuted adjacency rather than the semantic wildfire-risk ordering, the paper's central claim about intrinsic class separability would be falsified. Alternatively, compute the same diagnostics on a dataset with known class geometry where adjacent labels are semantically far apart and check whether the hierarchy inverts.

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

If this is right

  • If the hierarchy is intrinsic, confusion in four-class wildfire mapping is expected to concentrate between neighboring susceptibility levels, guiding where more training data or finer features are needed.
  • The grokking transition offers a concrete training-stage diagnostic: binary models suddenly generalize around 400–600 sweeps, so early stopping before that point would underestimate MPS performance.
  • The quantum-mask magnetization patterns indicate that binary low/high separation is a coarse projection of the full multiclass representation, suggesting a compositional structure that could transfer to other Mediterranean regions.
  • Level-resolved mixedness provides a new model-based way to probe label geometry, complementing class-wise accuracy and feature importance in environmental susceptibility studies.

Where Pith is reading between the lines

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

  • Editorial inference: the observed adjacency hierarchy may be an artifact of the ordinal label construction—since adjacent classes are defined by neighboring bins of a continuous fuel-load × danger product, they are likely more similar by design; a permutation test on class labels would separate semantic adjacency from label-construction adjacency.
  • Editorial inference: the same reduced-density-matrix mixedness diagnostics could be applied to any ordinal classification problem (e.g., drought severity, landslide risk) to check whether learned representations mirror the assumed ordinal scale.
  • Editorial inference: the grokking dynamics suggest a training strategy for tensor-network classifiers on geospatial data: use binary coarse tasks to prime the mask, then fine-tune on multiclass labels; the paper's magnetization correspondence hints this could speed convergence.

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 manuscript applies a Matrix Product State (MPS) tensor-network classifier to wildfire susceptibility mapping in the Gargano region, using 30 Boruta-selected AlphaEarth embedding features. It reports a binary task (class 0 vs. classes 1-3) with 87.07% accuracy and a four-class task with 53.01% accuracy, together with training curves that the authors interpret as a grokking transition. The advertised contribution is a set of level-resolved mixedness diagnostics - pairwise decoherence factors, triplet determinants, and conditional purities - which the authors claim show that the MPS naturally encodes a hierarchy of class distinguishability in which non-adjacent susceptibility classes are more separable than adjacent ones. The mathematical definitions of these diagnostics are coherent, and the internal consistency between diagnostics and class-wise accuracies is a useful observation. The main weakness is that the four-class labels are not independent ground truth: they are obtained by multiplying predicted fuel-load and ignition-danger maps and then reclassifying the product into ordered bins. The ordinal binning alone explains why adjacent classes are harder to separate, and no same-setup classical baseline is provided.

Significance. If established, the mixedness-diagnostic framework would be a novel and interpretable way to characterize class geometry in tensor-network classifiers, and the application to a real geospatial problem is a useful stress test of these models. The paper's mathematical core (Methods, Eqs. (1)-(8)) is sound and is a legitimate contribution to the toolkit for analyzing MPS classifiers. The grokking observation, while qualitative, is also of interest to the tensor-network-ML community. However, the central physical claim is currently under-supported. The label map is a modeled product reclassified into ordinal bins, and the reported RF benchmark is from a different pipeline; without a classical baseline on the same features and labels, the claimed 'intrinsic hierarchy' is confounded by label construction. With that baseline and a more cautious interpretation, the paper would be a reasonable contribution.

major comments (4)
  1. [Materials - Study area and wildfire reference data] The four-class label map is not independent ground truth. It is obtained by multiplying a predicted fine dead fuel load map by an ignition danger map and then reclassifying the product into ordered bins. Adjacent classes are adjacent intervals on a continuous modeled risk score. Any smooth classifier will find adjacent bins harder to separate than distant bins; the results in Fig. 5 are therefore what one expects from the label design. Since no classical baseline (e.g., logistic regression or random forest) is trained on the same 30 AlphaEarth features with the same four-class labels, the central claim that the MPS 'naturally encodes a hierarchy of class distinguishability' is not established. Please add baselines and compare confusion structure.
  2. [Results - Quantum masks mixedness (Fig. 5; Methods Eqs. (1), (6)-(8))] The mixedness measures are computed from the trained model's reduced density matrix. They describe the model's internal representation, not an independent measurement of wildfire susceptibility. The statement that 'the environment acquires more independent information for separating distant classes' equates the tensor-network environment with the physical environment, and this overreach should be tempered. The consistency among Gamma_ij, Delta_ijk, purity, and the class-wise accuracies in Table 1 is a useful sanity check, but it does not rule out the label-bin artifact. Show that an untrained MPS or a class-agnostic smooth model trained on the same inputs does not reproduce the same ordering.
  3. [Discussion] The paper cites the Random Forest model of ref. 9 as a strong benchmark (>95% accuracy), but that model is not a baseline: it uses different predictors, different label generation, and is not evaluated within the same train/test protocol. The abstract's phrase 'competitive classification accuracy' is therefore not supported by any in-paper comparison. Either train equivalent classical models on the same AlphaEarth features and four-class labels, or remove the 'competitive' claim. The MPS multiclass accuracy (53.01%, Table 1) is substantially lower than the cited RF accuracy, so the 'richer description' claim must be argued without relying on accuracy equivalence.
  4. [Results - grokking (Fig. 2)] The grokking transition is diagnosed by visual inspection of the training curves: a jump in test sensitivity around 400-600 sweeps. No quantitative criterion is given (e.g., a generalization-gap threshold or change-point test), and the number of runs is small (n=3). Since grokking is presented as a central result, the transition should be defined objectively; otherwise the claim is sensitive to smoothing and axis scaling choices.
minor comments (5)
  1. [Introduction] Typo: 'high-vulnerabil areas' should be 'high-vulnerability areas'.
  2. [Figure 1 caption] Typo: 'su-per-pixels' should be 'super-pixels'.
  3. [Methods - MPS classifier] The phrase 'The number of 30 features was selected' is ungrammatical; please rewrite as e.g. 'The set of 30 features was selected'. Also, 'the optimization of the tensor B' refers to an undefined symbol; presumably this is the MPS tensor W.
  4. [Table 1] Providing a confusion matrix for the multiclass task would directly support the adjacency claim and aid reproducibility.
  5. [References 14-15] References 14 and 15 are the authors' earlier papers on grokking in tensor networks. The Introduction should state explicitly what is reused from those works and what is new in the present contribution.

Circularity Check

0 steps flagged

No circular step found: the mixedness diagnostics characterize the trained MPS itself, and the ordinal-bin label confound is a modeling concern, not a definitional reduction.

full rationale

The paper's central claim is that level-resolved mixedness diagnostics (Γ_ij, Δ_ijk, conditional purities) computed from the trained MPS's label-space reduced density matrix reveal that non-adjacent susceptibility classes are more separable than adjacent ones. These diagnostics are explicit functions of the model's own parameters and are not used to fit any parameter; the claim is therefore about the learned representation, not a hidden reuse of the target. The four-class labels are admittedly constructed by multiplying a predicted fuel-load map with an ignition-danger map and reclassifying into ordered bins, which could explain the adjacency pattern independently of the MPS; but the paper states this construction openly and does not use the separability ordering as an input. No equation equates the reported hierarchy to the label-generation rule. The grokking interpretation cites prior work by overlapping authors (refs. 14, 15), but the transition is directly demonstrated in the training curves and those citations are terminological/supporting rather than load-bearing for the class-separability result. The lack of a classical baseline on the same features and labels is a benchmarking limitation, not circularity. Overall, the derivation chain is self-contained: model training, RDM construction, and mixedness diagnostics are all specified in the Methods, and the target result is not assumed in the inputs.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The core mathematical diagnostics are self-contained and correct, but the paper relies on external embedding quality and on label semantics that are only partially validated. Hyperparameters (feature count, χ, α) are fitted by hand on the same task, adding free parameters. No new physical entities are introduced.

free parameters (3)
  • Number of Boruta-selected features = 30
    Chosen by benchmarking 20, 30, and 40 features on the binary task, yielding test accuracies of 79.8%, 83.3%, and 82.9%.
  • Bond dimension χ = 1000 (binary), 500 (multiclass)
    Hand-set for each task; controls MPS capacity and likely influences grokking dynamics.
  • Learning rate α = 5e-3 (binary), 7e-4 (multiclass)
    Hand-set differently per task; confounds comparison of binary vs multiclass training dynamics.
axioms (5)
  • domain assumption The four-class susceptibility labels are valid ground truth for wildfire susceptibility.
    Labels are the product of two modeled maps (fuel load × ignition danger) reclassified into bins; no fire-occurrence validation is provided in this paper.
  • domain assumption AlphaEarth embeddings contain sufficient environmental signal for wildfire susceptibility.
    Relies on ref 4; no ablation against manually constructed predictors is performed here.
  • domain assumption Adjacent susceptibility classes are semantically 'neighboring' levels, so separability between non-adjacent classes is the meaningful finding.
    This ordinal interpretation is what turns the RDM diagnostics into a claim about class hierarchy.
  • domain assumption Sampling from super-pixels removes spatial autocorrelation.
    Assumes super-pixels are independent groups; no variogram or spatial autocorrelation test is reported.
  • standard math MSE amplitude training of an MPS is a valid classification procedure.
    Follows prior tensor-network ML (Stoudenmire & Schwab), but amplitude scores are not normalized probabilities.

pith-pipeline@v1.3.0-alltime-deepseek · 12499 in / 11789 out tokens · 109785 ms · 2026-08-01T12:32:17.541965+00:00 · methodology

0 comments
read the original abstract

A quantum-inspired tensor network framework for wildfire susceptibility classification in the Gargano region is introduced, leveraging AlphaEarth embeddings and Matrix Product State models. The approach combines scalable geospatial representations with an interpretable quantum mask, enabling both binary and multiclass classification of wildfire susceptibility. Beyond predictive performance, the study reveals a pronounced grokking transition in the binary case and provides a detailed analysis of inter-class confusion in the multiclass setting. By introducing level-resolved mixedness diagnostics based on reduced density matrices, we show that the MPS classifier naturally encodes a hierarchy of class distinguishability, with non-adjacent categories becoming more separable than neighboring ones. These results demonstrate that tensor network models not only achieve competitive classification accuracy but also offer a physically grounded framework to quantify and interpret class separability in complex environmental datasets.

Figures

Figures reproduced from arXiv: 2607.19503 by Alessandra Costantino, Alessandro Fania, Alfonso Monaco, Davide D' Al\`o, Domenico Pomarico, Ester Pantaleo, Francesco Giordano, Gabriel Ramirez Sanchez, Loredana Bellantuono, Mario Elia, Nicola Amoroso, Niloofar Kheirkhahan, Raffaele Lafortezza, Roberto Bellotti, Sabina Tangaro.

Figure 1
Figure 1. Figure 1: Study area and workflow for wildfire classification in the Gargano region. (a) Location of the study area within southern Italy and corresponding four-class wildfire susceptibility map at pixel resolution. (b) Schematic pipeline illustrating the group-fold sampling of embedding features from graded color super-pixels, binary (0 vs 1,2,3) and multiclass formulations, feature selection, MPS-based quantum mas… view at source ↗
Figure 2
Figure 2. Figure 2: Evolution of training and test performance of the MPS classifier as a function of training sweeps. The left panel is referred to binary classification (class 0 vs classes 1–3), the right panel to the four-class classification. Solid lines denote the mean values, while shaded regions represent the corresponding standard deviation computed over three independent runs. frameworks. The use of a regular grid en… view at source ↗
Figure 3
Figure 3. Figure 3: Magnetization dynamics of the MPS quantum mask for the binary classification problem, class 0 (left panel) vs classes 1–3 (right panel), showing the evolution of Pauli σZ expectation values across feature sites and training sweeps. Average magnetization is computed over three independent runs; the standard deviation exhibits a single peak of approximately 7×10−2 , observed only in the class-0 mask on the l… view at source ↗
Figure 4
Figure 4. Figure 4: Class-wise magnetization patterns for the four-class classification problem, illustrating the evolution of σZ expectation values for each susceptibility level. Average magnetization is computed over three independent runs; the standard deviation exhibits a single peak of approximately 8×10−2 , observed only in the class-0 mask near the grokking transition, followed by a rapid decay toward negligible values… view at source ↗
Figure 5
Figure 5. Figure 5: Evolution during four-class classification training of level-resolved mixedness metrics during MPS training. Panels (a–d) report pairwise decoherence factors Γi j, normalized triplet determinants ∆i jk, and conditional purities for pairs and triplets of susceptibility levels. Solid lines denote the mean values, while shaded regions represent the corresponding standard deviation computed over three independ… view at source ↗
Figure 6
Figure 6. Figure 6: Diagrammatics for MPS classifier quantities. The qubit encoding is represented with pink circles, in contraction to define the predictor shown in panel (a), where the variational tensor W is expressed as a MPS in the mixed-canonical form. The exploitation of an orthogonality center Λ enables efficient tensor contractions, for the pictorial purpose of a N = 4 lattice; (b) the reduced density matrix in label… view at source ↗

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Works this paper leans on

27 extracted references · 10 canonical work pages

  1. [1]

    Sebastianelli, A.et al.Quantum machine learning for earth observation: A review and future prospects.IEEE Geosci. Remote. Sens. Mag.14, 474–494, DOI: 10.1109/MGRS.2026.3676829 (2026)

    arXiv 2026

  2. [2]

    P., Ullo, S

    Sebastianelli, A., Del Rosso, M. P., Ullo, S. L. & Gamba, P. On quantum hyperparameters selection in hybrid classifiers for earth observation data.IEEE Geosci. Remote. Sens. Lett.20, 8501005, DOI: 10.1109/LGRS.2023.3308105 (2023)

    arXiv 2023

  3. [3]

    A., Spiller, D., Le Saux, B

    Sebastianelli, A., Zaidenberg, D. A., Spiller, D., Le Saux, B. & Ullo, S. On circuit-based hybrid quantum neural networks for remote sensing imagery classification.IEEE J. Sel. Top. Appl. Earth Obs. Remote. Sens.15, 565–580, DOI: 10.1109/JSTARS.2021.3134785 (2022)

    arXiv 2021

  4. [4]

    F.et al.AlphaEarth Foundations: An embedding field model for accurate and efficient global mapping from sparse label data.arXiv preprint arXiv:2507.22291(2025)

    Brown, C. F.et al.AlphaEarth Foundations: An embedding field model for accurate and efficient global mapping from sparse label data.arXiv preprint arXiv:2507.22291(2025)

    Pith/arXiv arXiv 2025

  5. [5]

    H., Cutugno, M., Robustelli, U

    Khachoo, Y . H., Cutugno, M., Robustelli, U. & Pugliano, G. Google earth engine since 2022: A structured bibliometric review of geoai-driven trends and applications.Sustainability18, 6241, DOI: 10.3390/su18126241 (2026)

    work page doi:10.3390/su18126241 2022

  6. [6]

    Atmospheric Sci.6, DOI: 10.1038/s41612-023-00423-1 (2023)

    Urdiales-Flores, D.et al.Drivers of accelerated warming in mediterranean climate-type regions.npj Clim. Atmospheric Sci.6, DOI: 10.1038/s41612-023-00423-1 (2023)

    work page doi:10.1038/s41612-023-00423-1 2023

  7. [7]

    Reports14, DOI: 10.1038/ s41598-024-80139-1 (2024)

    Lazoglou, G.et al.Identification of climate change hotspots in the mediterranean.Sci. Reports14, DOI: 10.1038/ s41598-024-80139-1 (2024)

    2024

  8. [8]

    Reports12, 16349, DOI: 10.1038/s41598-022-20347-9 (2022)

    Cilli, R.et al.Explainable artificial intelligence (xai) detects wildfire occurrence in the mediterranean countries of southern europe.Sci. Reports12, 16349, DOI: 10.1038/s41598-022-20347-9 (2022)

    doi:10.1038/s41598-022-20347-9 2022

  9. [9]

    ReportsDOI: 10.1038/s41598-026-52139-w (2026)

    Lafortezza, R.et al.Leveraging AI and multi-source satellite imagery for multi-hazard monitoring and susceptibility mapping in urban areas.Sci. ReportsDOI: 10.1038/s41598-026-52139-w (2026)

    work page doi:10.1038/s41598-026-52139-w 2026

  10. [10]

    D’Este, M.et al.Machine learning techniques for fine dead fuel load estimation using multi-source remote sensing data. Remote. Sens.13, 1658, DOI: 10.3390/rs13091658 (2021)

    work page doi:10.3390/rs13091658 2021

  11. [11]

    Elia, M.et al.Estimating the probability of wildfire occurrence in mediterranean landscapes using artificial neural networks. Environ. Impact Assess. Rev.85, 106474, DOI: 10.1016/j.eiar.2020.106474 (2020)

    arXiv 2020

  12. [12]

    & Vincent, P

    Bengio, Y ., Courville, A. & Vincent, P. Representation learning: A review and new perspectives.IEEE Trans. Pattern Anal. Mach. Intell.35, 1798–1828, DOI: 10.1109/TPAMI.2013.50 (2013)

    doi:10.1109/tpami.2013.50 2013

  13. [13]

    & Rudnicki, W

    Kursa, M., Jankowski, A. & Rudnicki, W. Boruta - a system for feature selection.Fundam. Inform.101, 271–285, DOI: 10.3233/FI-2010-288 (2010). 14.Pomarico, D.et al.Grokking as an entanglement transition in tensor network machine learning (2025). 2503.10483

    Pith/arXiv arXiv 2010

  14. [15]

    Pomarico, D.et al.Transfer entropy and O-information to detect grokking in tensor network multi-class classification problems.Technologies13, DOI: 10.3390/technologies13100438 (2025)

    work page doi:10.3390/technologies13100438 2025

  15. [16]

    & Doerr, S

    Shakesby, R. & Doerr, S. Wildfire as a hydrological and geomorphological agent.Earth-Science Rev.74, 269–307, DOI: https://doi.org/10.1016/j.earscirev.2005.10.006 (2006)

    work page doi:10.1016/j.earscirev.2005.10.006 2005

  16. [17]

    A., Shakesby, R

    Moody, J. A., Shakesby, R. A., Robichaud, P. R., Cannon, S. H. & Martin, D. A. Current research issues related to post- wildfire runoff and erosion processes.Earth-Science Rev.122, 10–37, DOI: https://doi.org/10.1016/j.earscirev.2013.03.004 (2013)

    work page doi:10.1016/j.earscirev.2013.03.004 2013

  17. [18]

    Zingaro, M.et al.Assessing the impact of vegetation cover changes and post-fire effects through an enhanced sediment flow connectivity index (sfci).CATENA247, 108474, DOI: https://doi.org/10.1016/j.catena.2024.108474 (2024)

    arXiv 2024

  18. [19]

    Gill, J. C. & Malamud, B. D. Reviewing and visualizing the interactions of natural hazards.Rev. Geophys.52, 680–722, DOI: https://doi.org/10.1002/2013RG000445 (2014). https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1002/2013RG000445. 13/14

    work page doi:10.1002/2013rg000445 2014

  19. [20]

    Gill, J. C. & Malamud, B. D. Hazard interactions and interaction networks (cascades) within multi-hazard methodologies. Earth Syst. Dyn.7, 659–679, DOI: 10.5194/esd-7-659-2016 (2016)

    work page doi:10.5194/esd-7-659-2016 2016

  20. [21]

    Tucker, L. R. Some mathematical notes on three-mode factor analysis.Psychometrika31, 279–311, DOI: 10.1007/ BF02289464 (1966)

    1966

  21. [22]

    Hackbusch, W.Tensor Spaces and Numerical Tensor Calculus(Springer Series in Computational Mathematics, Switzerland, 2019)

    2019

  22. [23]

    & Vandewalle, J

    De Lathauwer, L., De Moor, B. & Vandewalle, J. A multilinear singular value decomposition.SIAM J. on Matrix Analysis Appl.21, 1253–1278, DOI: 10.1137/S0895479896305696 (2000). 24.Oseledets, I. V . Tensor-train decomposition.SIAM J. on Sci. Comput.33, 2295–2317, DOI: 10.1137/090752286 (2011)

    doi:10.1137/s0895479896305696 2000

  23. [25]

    White, S. R. Density matrix formulation for quantum renormalization groups.Phys. Rev. Lett.69, 2863–2866, DOI: 10.1103/PhysRevLett.69.2863 (1992)

    doi:10.1103/physrevlett.69.2863 1992

  24. [26]

    The density-matrix renormalization group.Rev

    Schollwöck, U. The density-matrix renormalization group.Rev. Mod. Phys.77, 259–315, DOI: 10.1103/RevModPhys.77. 259 (2005)

    doi:10.1103/revmodphys.77 2005

  25. [27]

    The density-matrix renormalization group in the age of matrix product states.Annals Phys.326, 96–192, DOI: https://doi.org/10.1016/j.aop.2010.09.012 (2011)

    Schollwöck, U. The density-matrix renormalization group in the age of matrix product states.Annals Phys.326, 96–192, DOI: https://doi.org/10.1016/j.aop.2010.09.012 (2011). January 2011 Special Issue. 28.Stoudenmire, E. M. & Schwab, D. J. Supervised learning with quantum-inspired tensor networks (2017). 1605.05775

    Pith/arXiv arXiv 2010

  26. [29]

    Wiersema, R., Zhou, C., Carrasquilla, J. F. & Kim, Y . B. Measurement-induced entanglement phase transitions in variational quantum circuits.SciPost Phys.14, 147, DOI: 10.21468/SciPostPhys.14.6.147 (2023)

    work page doi:10.21468/scipostphys.14.6.147 2023

  27. [30]

    Budget MUR - Dipartimenti di Eccellenza 2023 - 2027

    Horn, R. A. & Johnson, C. R.Matrix Analysis(Cambridge University Press, Cambridge ; New York, NY , 2012), 2nd edn. 31.Bourbaki, N.Elements of Mathematics, Algebra I(Springer-Verlag, Berlin, Heidelberg, 1989). Funding Declaration Authors were supported by the Italian funding within the “Budget MUR - Dipartimenti di Eccellenza 2023 - 2027” (Law 232, 11 Dece...

    2012