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REVIEW 2 major objections 4 minor 63 references

Local reduced density matrices of one to four sites suffice to classify topological phases, including SPT phases, via a quantum kernel fed to a classical SVM.

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2026-07-14 10:10 UTC pith:E5OZXBKK

load-bearing objection Solid numerical demo that local RDM kernels (NA=1–4) recover full SPT phase diagrams of two standard 1D models, with clean size generalization; incremental but useful for experiment. the 2 major comments →

arxiv 2607.10656 v1 pith:E5OZXBKK submitted 2026-07-12 quant-ph cond-mat.stat-mechcond-mat.str-elcs.LG

Learning Topological Quantum Phases from Limited Subsystems

Mehran Khosrojerdi , Sougato Bose , Alessandro Cuccoli , Paola Verrucchi , Abolfazl Bayat , Leonardo Banchi This is my paper
classification quant-ph cond-mat.stat-mechcond-mat.str-elcs.LG
keywords topological quantum phasessymmetry-protected topological phasesreduced density matricesquantum kernelssupport vector machinescluster-Ising modelHaldane chainmatrix product states
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.

Characterizing topological phases of quantum matter normally requires measuring non-local string order parameters across an entire many-body system, which is hard in experiment. This paper shows that those global phases can still be identified from tiny accessible fragments. The authors construct a quantum kernel from the reduced density matrices of one-to-four contiguous sites, feed that kernel into a classical support-vector machine, and recover the full phase diagrams of two paradigmatic one-dimensional models that each contain a symmetry-protected topological phase. The same classifier, trained on chains of moderate length, continues to work when applied to longer chains. The practical message is that the local reduced density matrix already carries enough information to distinguish topological phases, so experimental access need not be global.

Core claim

Reduced density matrices of as few as one to four sites retain the signatures needed to classify all phases—including the symmetry-protected topological phases—of the generalized cluster-Ising spin-1/2 chain and the anisotropic Haldane spin-1 chain when those matrices are used as a quantum kernel for a classical support-vector machine. The classifier trained on chains of length 31 generalizes without retraining to chains of length 51.

What carries the argument

The quantum kernel K(x_n, x_m) = Tr[ρ_A(x_n) ρ_A(x_m)], built from the reduced density matrices of a small contiguous block A; this kernel is the sole input to a classical multi-class SVM that learns the decision boundaries separating the phases.

Load-bearing premise

The phase labels used for supervised training must already be correct, because they are taken from earlier infinite-size calculations rather than being discovered by the method itself.

What would settle it

Recompute the same local kernels on the same two models but deliberately mis-label a subset of the training points that lie deep inside each phase; if test accuracy remains high against the published diagrams, the method is merely reproducing the geometry of the training labels rather than recognizing phases.

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

If this is right

  • Phase identification of SPT and other topological phases becomes possible with only local experimental access to a few contiguous sites.
  • The same trained classifier can be reused on longer chains without recomputing global ground states or string order parameters.
  • Kernel entries can be estimated by swap tests, single-triplet measurements, classical shadows or small-scale tomography, all of which are within reach of present-day hardware.
  • Once the SVM coefficients are known, the decision observable itself can be rewritten as a short Pauli string that is measured directly on the subsystem.

Where Pith is reading between the lines

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

  • If the local-kernel method continues to work for two-dimensional or higher-order topological phases, full-system string-order measurements may become unnecessary for routine phase classification.
  • The rapid size-convergence of the reduced density matrix suggests that training data can be generated cheaply on small systems and then deployed on experimentally relevant lengths.
  • The success with a single central site for the Haldane phase implies that the space of one-site density matrices already separates the topological region from its neighbors by exclusion.

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

2 major / 4 minor

Summary. The paper proposes a supervised SVM classifier that uses a quantum kernel K(x_n, x_m) = Tr[ρ_A(x_n) ρ_A(x_m)] built from reduced density matrices of small contiguous blocks A (N_A = 1–4 sites) extracted from MPS ground states. It is applied to reconstruct the known phase diagrams of the generalized cluster-Ising spin-1/2 chain and the anisotropic Haldane spin-1 chain, both of which contain SPT phases. High classification accuracy is reported when A is placed at the center (and, with larger N_A, at the edge), and the classifier trained on L = 31 chains generalizes to L = 51 chains. The authors conclude that local RDMs already encode enough information to identify global topological phases once phase labels are supplied.

Significance. If the numerical evidence holds, the result is practically useful: it shows that experimentally accessible local tomography or swap-test kernels can replace non-local string-order measurements for phase identification in 1-D SPT systems. The size-generalization property (training on moderate L, testing on longer L) further lowers the computational and experimental overhead. Strengths include the transparent kernel construction (Eq. 2), the use of standard MPS libraries with stated bond dimensions, and the clear visual and quantitative benchmarks (Figs. 3–7) on two well-studied models. The work does not claim unsupervised discovery; its contribution is the demonstration that local kernels suffice for supervised classification of topological diagrams.

major comments (2)
  1. [Section II, Figs. 2–6] Section II and Figs. 2–6: the training labels are taken from previously published infinite-size DMRG/MPS phase boundaries (Refs. [23, 38, 58]). While this is not circular for a supervised method, the manuscript never quantifies how sensitive the reported accuracies (Fig. 7) are to small shifts or uncertainties in those external boundaries. A short robustness check—e.g., randomly flipping a few percent of training labels near the critical lines—would strengthen the claim that the local kernel truly separates the phases rather than merely interpolating the supplied geometry.
  2. [Section III.C, Fig. 7] Section III.C and Fig. 7: accuracy is reported as a single number per N_A without error bars, cross-validation folds, or variation over different random draws of the training set. Because the number and precise locations of the colored training points in Fig. 2 are not specified beyond “randomly chosen,” it is impossible to assess statistical reliability or reproducibility of the high-accuracy claims (especially the single-site SPT recognition).
minor comments (4)
  1. [Throughout] Several typos appear in the text: “dge of che chain” (p. 7), “N´ eel” accents, “T=0 properties” (p. 3), and “support vector machines with quantum kernels” header capitalization. A careful proof-reading pass is needed.
  2. [Section II, Fig. 1] Fig. 1 caption and surrounding text: the mixed-canonical MPS construction is standard, yet the precise truncation of the bond index γ to χ is mentioned only in passing; a one-sentence statement of the discarded weight would help readers judge the quality of the ρ_A used for the kernels.
  3. [Section II] The experimental estimation routes (swap test, shadows, tomography) are listed but never quantified; even a rough shot-count estimate for N_A = 4 would make the “efficiently estimated experimentally” claim more concrete.
  4. [Figs. 5–6] In the Haldane-model panels the color scale for the three Néel phases is hard to distinguish in grayscale printouts; a different marker style or hatching would improve accessibility.

Circularity Check

0 steps flagged

No definitional or fitted circularity; supervised local-kernel SVM reproduces externally labeled phase diagrams without reducing claims to inputs by construction.

full rationale

The paper's central derivation is a standard supervised pipeline: MPS ground states of the GCI and anisotropic Haldane Hamiltonians yield reduced density matrices ho_A of 1–4 sites; the quantum kernel K(x_n,x_m)=Tr[ ho_A(x_n) ho_A(x_m)] is fed to a classical SVM (C=1 default) whose training labels are taken from previously published infinite-size DMRG/MPS phase diagrams (Refs. [23,38,58] and Fig. 2). Test accuracy is then measured against a regular grid of the same external labels (Figs. 3–7). Nothing in Eqs. (2)–(6) or the optimization (4) forces the decision boundaries by definition or by a fit that is later re-labeled a prediction; the kernel is an independent similarity measure on local RDMs, and size generalization (L=31 train o L=51 test) follows from rapid convergence of ho_A, which is an empirical observation rather than a tautology. Self-citations to the authors’ prior ML works ([34,47]) appear only as additional confirmations of the known diagrams and are not load-bearing for the local-kernel claim. The supervised character is openly stated in the conclusions and does not constitute circularity under the stated criteria. Score 1 reflects only the mild, non-load-bearing use of the authors’ own earlier classifications as supporting literature.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

The central claim rests on standard quantum many-body tools (MPS, reduced density matrices, quantum kernels, SVM theory) plus the externally established phase diagrams of the two models. Free parameters are the usual numerical cut-offs and the SVM hyper-parameter; no new physical entities are postulated.

free parameters (3)
  • SVM regularization C = 1
    Fixed to default value C=1; controls the soft-margin trade-off and therefore the decision boundaries.
  • MPS bond dimension χ = 150 / 50
    Truncation parameter for ground-state accuracy; set to 150 (cluster-Ising) and 50 (Haldane).
  • Number and placement of training points
    Hand-chosen scattered points that cover the three (or more) phases; their density and locations affect the learned hyperplanes.
axioms (4)
  • domain assumption Ground states of the two Hamiltonians are accurately represented by finite-bond MPS in the mixed-canonical form.
    Invoked throughout Section II and used to extract all RDMs; standard but not proved for the chosen χ.
  • domain assumption The phase diagrams of the GCI and anisotropic Haldane models are those previously obtained by infinite-size DMRG/MPS (Figs. 2a,b).
    Supplies the ground-truth labels for both training and accuracy evaluation.
  • standard math The representer theorem allows the SVM weight operator W to be expanded solely in the training RDMs, yielding a convex dual problem whose solution depends only on the kernel matrix.
    Standard SVM theory (Eqs. 3–5); used to justify that only the kernel need be computed.
  • domain assumption Reduced density matrices of fixed block size NA converge rapidly with total chain length L, so that a classifier trained at L=31 remains valid at L=51.
    Empirical observation used to claim size-generalizability (Figs. 4,6 and text of Sec. III).

pith-pipeline@v1.1.0-grok45 · 17504 in / 2580 out tokens · 25417 ms · 2026-07-14T10:10:28.335472+00:00 · methodology

0 comments
read the original abstract

Characterizing quantum topological phases requires measuring non-local string order parameters, demanding access to the full system, which is often experimentally unfeasible. In this work, we introduce a data-efficient supervised learning framework that circumvents this limitation by recognizing quantum phases from small subsystems. Our protocol utilizes a quantum kernel constructed from the reduced density matrices of these subsystems, which can be efficiently estimated experimentally. We benchmark our framework with the classification of the phase diagrams of two spin models on one-dimensional lattices, namely the generalized cluster-Ising spin-1/2 chain and the anisotropic Haldane spin-1 chain. Remarkably, our approach achieves high accuracy in phase classification when operations are limited to as few as one to four sites, and it also generalizes to longer chains even when trained on moderate system sizes. These findings demonstrate that local reduced density matrices preserve vital signatures of global topological phases, offering a practical route to characterize rich phase diagrams of quantum many-body systems.

Figures

Figures reproduced from arXiv: 2607.10656 by Abolfazl Bayat, Alessandro Cuccoli, Leonardo Banchi, Mehran Khosrojerdi, Paola Verrucchi, Sougato Bose.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Bipartite MPS in canonical form. The chain is split into a block of adjacent sites [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Phase diagram of the Generalized Cluster-Ising model, Eq. (7): phase-boundary lines are the results of infinite [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Phase diagram reconstruction of the Generalized Cluster-Ising model based on reduced density matrices for blocks of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Reconstruction of the phase diagram of the Generalized Cluster-Ising model as in Fig. 3; while a chain of 51 sites is [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Phase diagram reconstruction of the anisotropic Haldane model based on reduced density matrices for blocks of [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Reconstruction of the phase diagram of the anisotropic Haldane model as in Fig. 5; while a chain of 51 sites is still [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Accuracy in phase classification, namely fraction of correctly classified test points versus number of kept sites, for (a) [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗

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Reference graph

Works this paper leans on

63 extracted references · 3 linked inside Pith

  1. [1]

    Sachdev, Quantum phase transitions, Physics world12, 33 (1999)

    S. Sachdev, Quantum phase transitions, Physics world12, 33 (1999)

    1999

  2. [2]

    Vojta, Quantum phase transitions, Reports on Progress in Physics66, 2069 (2003)

    M. Vojta, Quantum phase transitions, Reports on Progress in Physics66, 2069 (2003)

    2069

  3. [3]

    Mendes-Santos, G

    T. Mendes-Santos, G. Giudici, M. Dalmonte, and M. Rajabpour, Entanglement hamiltonian of quantum critical chains and conformal field theories, Physical Review B100, 155122 (2019)

    2019

  4. [4]

    Calabrese and A

    P. Calabrese and A. Lefevre, Entanglement spectrum in one-dimensional systems, Physical Review A—Atomic, Molecular, and Optical Physics78, 032329 (2008)

    2008

  5. [5]

    Calabrese and J

    P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, Journal of statistical mechanics: theory and experiment2004, P06002 (2004)

    2004

  6. [6]

    Calabrese, J

    P. Calabrese, J. Cardy, and B. Doyon, Entanglement entropy in extended quantum systems, J. Phys. A42, 500301 (2009)

    2009

  7. [7]

    Bayat, P

    A. Bayat, P. Sodano, and S. Bose, Negativity as the entanglement measure to probe the kondo regime in the spin-chain kondo model, Physical Review B—Condensed Matter and Materials Physics81, 064429 (2010)

    2010

  8. [8]

    R. A. Santos, V. Korepin, and S. Bose, Negativity for two blocks in the one-dimensional spin-1 affleck-kennedy-lieb-tasaki model, Physical Review A—Atomic, Molecular, and Optical Physics84, 062307 (2011)

    2011

  9. [9]

    Calabrese, J

    P. Calabrese, J. Cardy, and E. Tonni, Entanglement negativity in quantum field theory, Physical review letters109, 130502 (2012)

    2012

  10. [10]

    Ruggiero, V

    P. Ruggiero, V. Alba, and P. Calabrese, Negativity spectrum of one-dimensional conformal field theories, Physical Review B94, 195121 (2016)

    2016

  11. [11]

    W. Chen, K. Hida, and B. Sanctuary, Ground-state phase diagram of s= 1 xxz chains with uniaxial single-ion-type anisotropy, Physical Review B67, 104401 (2003)

    2003

  12. [12]

    Langari, F

    A. Langari, F. Pollmann, and M. Siahatgar, Ground-state fidelity of the spin-1 heisenberg chain with single ion anisotropy: quantum renormalization group and exact diagonalization approaches, Journal of Physics: Condensed Matter25, 406002 (2013)

    2013

  13. [13]

    Ejima and H

    S. Ejima and H. Fehske, Comparative density-matrix renormalization group study of symmetry-protected topological phases in spin-1 chain and bose-hubbard models, Physical Review B91, 045121 (2015)

    2015

  14. [14]

    Yu and W.-L

    X.-J. Yu and W.-L. Li, Quantum phase transition between topologically distinct quantum critical points, arXiv preprint arXiv:2403.03716 (2024)

    Pith/arXiv arXiv 2024

  15. [15]

    N. U. K¨ oyl¨ uo˘ glu, S. Majumder, M. Amico, S. Mostame, E. van den Berg, M. Rajabpour, Z. Minev, and K. Najafi, Measuring central charge on a universal quantum processor, Nature Communications (2026)

    2026

  16. [16]

    A. F. Albuquerque, C. J. Hamer, and J. Oitmaa, Quantum phase diagram and excitations for the one-dimensional s= 1 heisenberg antiferromagnet with single-ion anisotropy, Physical Review B—Condensed Matter and Materials Physics79, 054412 (2009)

    2009

  17. [17]

    Smacchia, L

    P. Smacchia, L. Amico, P. Facchi, R. Fazio, G. Florio, S. Pascazio, and V. Vedral, Statistical mechanics of the cluster ising model, Physical Review A—Atomic, Molecular, and Optical Physics84, 022304 (2011)

    2011

  18. [18]

    W. Son, L. Amico, R. Fazio, A. Hamma, S. Pascazio, and V. Vedral, Quantum phase transition between cluster and antiferromagnetic states, Europhysics Letters95, 50001 (2011)

    2011

  19. [19]

    P´ erez-Garc´ ıa, M

    D. P´ erez-Garc´ ıa, M. M. Wolf, M. Sanz, F. Verstraete, and J. I. Cirac, String order and symmetries in quantum spin lattices, Physical review letters100, 167202 (2008)

    2008

  20. [20]

    Pollmann, A

    F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, Entanglement spectrum of a topological phase in one dimension, Physical Review B—Condensed Matter and Materials Physics81, 064439 (2010)

    2010

  21. [21]

    T. Ohta, S. Tanaka, I. Danshita, and K. Totsuka, Topological and dynamical properties of a generalized cluster model in one dimension, Physical Review B93, 165423 (2016)

    2016

  22. [22]

    T. Ohta, S. Tanaka, I. Danshita, and K. Totsuka, Phase diagram and sweep dynamics of a one-dimensional generalized cluster model, Journal of the Physical Society of Japan84, 063001 (2015)

    2015

  23. [23]

    Tzeng, H

    Y.-C. Tzeng, H. Onishi, T. Okubo, and Y.-J. Kao, Quantum phase transitions driven by rhombic-type single-ion anisotropy in thes= 1 haldane chain, Phys. Rev. B96, 060404 (2017)

    2017

  24. [24]

    Tzeng and M.-F

    Y.-C. Tzeng and M.-F. Yang, Scaling properties of fidelity in the spin-1 anisotropic model, Physical Review A—Atomic, Molecular, and Optical Physics77, 012311 (2008). 10

    2008

  25. [25]

    D.-L. Deng, X. Li, and S. Das Sarma, Machine learning topological states, Phys. Rev. B96, 195145 (2017)

    2017

  26. [26]

    Lian, S.-T

    W. Lian, S.-T. Wang, S. Lu, Y. Huang, F. Wang, X. Yuan, W. Zhang, X. Ouyang, X. Wang, X. Huang, L. He, X. Chang, D.-L. Deng, and L. Duan, Machine learning topological phases with a solid-state quantum simulator, Phys. Rev. Lett.122, 210503 (2019)

    2019

  27. [27]

    Zhang, S

    H. Zhang, S. Jiang, X. Wang, W. Zhang, X. Huang, X. Ouyang, Y. Yu, Y. Liu, D.-L. Deng, and L.-M. Duan, Experimental demonstration of adversarial examples in learning topological phases, Nature communications13, 4993 (2022)

    2022

  28. [28]

    Zhang, W

    X. Zhang, W. Jiang, J. Deng, K. Wang, J. Chen, P. Zhang, W. Ren, H. Dong, S. Xu, Y. Gao,et al., Digital quantum simulation of floquet symmetry-protected topological phases, Nature607, 468 (2022)

    2022

  29. [29]

    Xiang, W

    L. Xiang, W. Jiang, Z. Bao, Z. Song, S. Xu, K. Wang, J. Chen, F. Jin, X. Zhu, Z. Zhu,et al., Long-lived topological time-crystalline order on a quantum processor, Nature Communications15, 8963 (2024)

    2024

  30. [30]

    Carrasquilla and R

    J. Carrasquilla and R. G. Melko, Machine learning phases of matter, Nature Physics13, 431 (2017)

    2017

  31. [31]

    Cristianini and J

    N. Cristianini and J. Shawe-Taylor,An Introduction to Support Vector Machines and Other Kernel-based Learning Methods (Cambridge University Press, 2000)

    2000

  32. [32]

    Schuld, Supervised quantum machine learning models are kernel methods, arXiv preprint arXiv:2101.11020 (2021)

    M. Schuld, Supervised quantum machine learning models are kernel methods, arXiv preprint arXiv:2101.11020 (2021)

    Pith/arXiv arXiv 2021

  33. [33]

    Huang, R

    H.-Y. Huang, R. Kueng, G. Torlai, V. V. Albert, and J. Preskill, Provably efficient machine learning for quantum many- body problems, Science377, eabk3333 (2022)

    2022

  34. [34]

    Khosrojerdi, A

    M. Khosrojerdi, A. Cuccoli, P. Verrucchi, and L. Banchi, Unsupervised learning to recognize quantum phases of matter, arXiv preprint arXiv:2510.14742 (2025)

    arXiv 2025

  35. [35]

    Y.-D. Wu, Y. Zhu, Y. Wang, and G. Chiribella, Learning quantum properties from short-range correlations using multi-task networks, Nature Communications15, 8796 (2024)

    2024

  36. [36]

    Y. Du, Y. Zhu, Y.-H. Zhang, M.-H. Hsieh, P. Rebentrost, W. Gao, Y.-D. Wu, J. Eisert, G. Chiribella, D. Tao,et al., Artificial intelligence for representing and characterizing quantum systems, arXiv preprint arXiv:2509.04923 (2025)

    Pith/arXiv arXiv 2025

  37. [37]

    Y.-D. Wu, Y. Zhu, G. Chiribella, and N. Liu, Efficient learning of continuous-variable quantum states, Phys. Rev. Res.6, 033280 (2024)

    2024

  38. [38]

    I. Cong, S. Choi, and M. D. Lukin, Quantum convolutional neural networks, Nature Physics15, 1273 (2019)

    2019

  39. [39]

    Q. Li, Y. Huang, X. Hou, Y. Li, X. Wang, and A. Bayat, Ensemble-learning error mitigation for variational quantum shallow-circuit classifiers, Physical Review Research6, 013027 (2024)

    2024

  40. [40]

    Lazzarin, D

    M. Lazzarin, D. E. Galli, and E. Prati, Multi-class quantum classifiers with tensor network circuits for quantum phase recognition, Physics Letters A434, 128056 (2022)

    2022

  41. [41]

    Gong, H.-L

    M. Gong, H.-L. Huang, S. Wang, C. Guo, S. Li, Y. Wu, Q. Zhu, Y. Zhao, S. Guo, H. Qian,et al., Quantum neuronal sensing of quantum many-body states on a 61-qubit programmable superconducting processor, Science Bulletin68, 906 (2023)

    2023

  42. [42]

    Mansuroglu, T

    R. Mansuroglu, T. Eckstein, L. N¨ utzel, S. A. Wilkinson, and M. J. Hartmann, Variational hamiltonian simulation for translational invariant systems via classical pre-processing, Quantum Science and Technology8, 025006 (2023)

    2023

  43. [43]

    M. S. Tepaske, D. Hahn, and D. J. Luitz, Optimal compression of quantum many-body time evolution operators into brickwall circuits, SciPost Physics14, 073 (2023)

    2023

  44. [44]

    Mc Keever and M

    C. Mc Keever and M. Lubasch, Classically optimized hamiltonian simulation, Physical review research5, 023146 (2023)

    2023

  45. [45]

    Lloyd, M

    S. Lloyd, M. Mohseni, and P. Rebentrost, Quantum principal component analysis, Nature physics10, 631 (2014)

    2014

  46. [46]

    Banchi, J

    L. Banchi, J. L. Pereira, S. T. Jose, and O. Simeone, Statistical complexity of quantum learning, Advanced Quantum Technologies8, 2300311 (2025)

    2025

  47. [47]

    Khosrojerdi, J

    M. Khosrojerdi, J. L. Pereira, A. Cuccoli, and L. Banchi, Learning to classify quantum phases of matter with a few measurements, Quantum Science and Technology10, 025006 (2025)

    2025

  48. [48]

    Parigi, M

    M. Parigi, M. Khosrojerdi, F. Caruso, and L. Banchi, Analysis of quantum image representations for supervised classifica- tion, AVS Quantum Science8(2026)

    2026

  49. [49]

    Sancho-Lorente, J

    T. Sancho-Lorente, J. Rom´ an-Roche, and D. Zueco, Quantum kernels to learn the phases of quantum matter, Physical Review A105, 042432 (2022)

    2022

  50. [50]

    Y. Wu, B. Wu, J. Wang, and X. Yuan, Quantum phase recognition via quantum kernel methods, Quantum7, 981 (2023)

    2023

  51. [51]

    J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nature communications9, 4812 (2018)

    2018

  52. [52]

    Banchi, A

    L. Banchi, A. Bayat, and S. Bose, Entanglement entropy scaling in solid-state spin arrays via capacitance measurements, Physical Review B94, 241117 (2016)

    2016

  53. [53]

    Huang, R

    H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measurements, Nature Physics16, 1050 (2020)

    2020

  54. [54]

    J. Haah, A. W. Harrow, Z. Ji, X. Wu, and N. Yu, Sample-optimal tomography of quantum states, IEEE Transactions on Information Theory63, 5628 (2017)

    2017

  55. [55]

    Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Annals of physics326, 96 (2011)

    U. Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Annals of physics326, 96 (2011)

    2011

  56. [56]

    Gray, quimb: A python package for quantum information and many-body calculations, Journal of Open Source Software 3, 819 (2018)

    J. Gray, quimb: A python package for quantum information and many-body calculations, Journal of Open Source Software 3, 819 (2018)

    2018

  57. [57]

    Pollmann and A

    F. Pollmann and A. M. Turner, Detection of symmetry-protected topological phases in one dimension, Physical Review B—Condensed Matter and Materials Physics86, 125441 (2012)

    2012

  58. [58]

    A. Sone, A. Tanji, and N. Yamamoto, Quantum inception score, Physical Review Research6, 033198 (2024)

    2024

  59. [59]

    J. Ren, Y. Wang, and W.-L. You, Quantum phase transitions in spin-1 xxz chains with rhombic single-ion anisotropy, Physical Review A97, 042318 (2018). 11

    2018

  60. [60]

    Nico-Katz and S

    A. Nico-Katz and S. Bose, Entanglement-complexity geometric measure, Physical Review Research5, 013041 (2023)

    2023

  61. [61]

    Den Nijs and K

    M. Den Nijs and K. Rommelse, Preroughening transitions in crystal surfaces and valence-bond phases in quantum spin chains, Physical Review B40, 4709 (1989)

    1989

  62. [62]

    Zauner, D

    V. Zauner, D. Draxler, L. Vanderstraeten, J. Haegeman, and F. Verstraete, Symmetry breaking and the geometry of reduced density matrices, New Journal of Physics18, 113033 (2016)

    2016

  63. [63]

    C. L. Henley and H. J. Changlani, Density-matrix based numerical methods for discovering order and correlations in interacting systems, Journal of Statistical Mechanics: Theory and Experiment2014, P11002 (2014)

    2014