pith. sign in

arxiv: 2503.17230 · v3 · pith:4Z3TO3QTnew · submitted 2025-03-21 · 🪐 quant-ph · cond-mat.dis-nn

Tensor Cross Interpolation of Purities in Quantum Many-Body Systems

Dmytro Kolisnyk , Raimel A. Medina , Romain Vasseur , Maksym Serbyn This is my paper

Pith reviewed 2026-05-22 22:38 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.dis-nn
keywords entanglement featuretensor cross interpolationmatrix product statepurityRenyi entropyquantum many-bodyeigenstatesone-dimensional systems
0
0 comments X

The pith

The entanglement feature of a quantum many-body state can be learned from polynomially many purity samples via tensor cross interpolation when it has finite matrix-product rank.

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

The paper establishes that the complete set of bipartite Renyi purities across all subregions, packaged into an entanglement feature, can be recovered without exponential cost. Treating this feature as amplitudes of a fictitious wave function that admits a low-bond-dimension matrix product state representation allows the tensor cross interpolation algorithm to reconstruct it from a polynomial number of measurements. Benchmarks on Haar-random and random MPS states confirm the expected scaling, and the method is then applied to eigenstates of one-dimensional spin chains to locate cases where the feature remains learnable. The resulting interpolated feature supports downstream tasks such as measuring distances between entanglement patterns and optimizing the ordering of physical sites.

Core claim

Assuming the entanglement feature is expressible as a finite-bond-dimension matrix product state, the tensor cross interpolation algorithm reconstructs all subregion purities using only a polynomial number of samples in the number of degrees of freedom, as verified on random states and demonstrated on eigenstates of various one-dimensional quantum Hamiltonians.

What carries the argument

Tensor cross interpolation (TCI) algorithm applied to the entanglement feature represented as a matrix product state of finite bond dimension.

If this is right

  • All pairwise distances between different entanglement patterns become computable from the learned feature.
  • The optimal one-dimensional ordering of physical indices for a given state can be found by minimizing a cost derived from the interpolated purities.
  • Eigenstates of one-dimensional quantum systems can be classified according to whether their entanglement features are efficiently learnable.
  • Full tomography of the entanglement feature becomes feasible for states obeying the finite-bond-dimension assumption.

Where Pith is reading between the lines

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

  • The same interpolation approach could be tested on states whose entanglement features arise from integrable or weakly entangled dynamics beyond one dimension.
  • If many physically relevant states satisfy the finite-bond-dimension condition, purity-based diagnostics could replace full state tomography in quantum simulation workflows.
  • The learned feature might serve as input for machine-learning models that predict dynamical properties from static entanglement data.

Load-bearing premise

The entanglement feature must admit a representation as a matrix product state with finite bond dimension.

What would settle it

A state whose entanglement feature requires bond dimension that grows exponentially with system size, forcing TCI sample complexity to become exponential.

Figures

Figures reproduced from arXiv: 2503.17230 by Dmytro Kolisnyk, Maksym Serbyn, Raimel A. Medina, Romain Vasseur.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) All possible bipartite purities of an entangled [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Schematic depiction of matrix cross interpola [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Tensor cross interpolation of Haar random state en [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Construction of the MPS form of the EF in [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Average bond dimension of entanglement feature [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Three qualitatively different behaviors of average bond dimension of entanglement feature of (a) highly excited MBL [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Entanglement map illustrating the entanglement [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (a) Seemingly volume-law randomly shuffled MPS can be reordered into a locally clustered area-law state. (b) [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. EF complexity of (a) Fredkin state, (b) colorless Motzkin state, (c)-(d) two-color Motzkin state considering standard [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Number of queries to the purity function used by TCI for the Mode-II reconstruction of (a) highly excited eigenstates of [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Raw statistics of the entanglement feature distances. System size is fixed to [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
read the original abstract

A defining feature of quantum many-body systems is the exponential scaling of the Hilbert space with the number of degrees of freedom. This exponential complexity na\"ively renders a complete state characterization, for instance via the complete set of bipartite Renyi entropies for all disjoint regions, a challenging task. Recently, a compact way of storing subregions' purities by encoding them as amplitudes of a fictitious quantum wave function, known as entanglement feature, was proposed. Notably, the entanglement feature can be a simple object even for highly entangled quantum states. However the complexity and practical usage of the entanglement feature for general quantum states has not been explored. In this work, we demonstrate that the entanglement feature can be efficiently learned using only a polynomial amount of samples in the number of degrees of freedom through the so-called tensor cross interpolation (TCI) algorithm, assuming it is expressible as a finite bond dimension MPS. We benchmark this learning process on Haar and random MPS states, confirming analytic expectations. Applying the TCI algorithm to quantum eigenstates of various one dimensional quantum systems, we identify cases where eigenstates have entanglement feature learnable with TCI. We conclude with possible applications of the learned entanglement feature, such as quantifying the distance between different entanglement patterns and finding the optimal one-dimensional ordering of physical indices in a given state, highlighting the potential utility of the proposed purity interpolation method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that the entanglement feature (encoding of bipartite Rényi purities as amplitudes of a fictitious wave function) of quantum many-body states can be reconstructed using tensor cross interpolation (TCI) with only polynomially many samples in the number of degrees of freedom, provided the feature admits a finite-bond-dimension MPS representation. Benchmarks on Haar-random states and random MPS states are shown to match analytic expectations; the method is then applied to eigenstates of various 1D Hamiltonians to identify cases where TCI converges, with suggested applications to entanglement-pattern distances and optimal index ordering.

Significance. If the finite-bond-dimension assumption holds for physically relevant states, the approach supplies a concrete, sample-efficient route to characterizing full sets of subregion purities without exponential resources. The explicit benchmarks on analytically tractable ensembles and the conditional framing of the eigenstate results constitute clear strengths; the work thereby opens a practical path toward comparing entanglement structures across states or optimizing tensor-network representations.

major comments (2)
  1. [Applications to eigenstates] Applications section: the identification of 'cases where eigenstates have entanglement feature learnable with TCI' reports only successful convergences and supplies neither reconstruction error bars, nor statistics on the fraction of eigenstates or system sizes for which TCI fails to converge, nor any scan of required bond dimension versus disorder or system size; this omission leaves the prevalence of the finite-bond-dimension property unquantified and weakens assessment of practical utility beyond the benchmark ensembles.
  2. [Benchmarks] Benchmark and method sections: while the TCI reconstruction matches analytic expectations on Haar-random and random-MPS states (where the MPS assumption holds by construction), the manuscript contains no explicit tests or failure-mode analysis on states deliberately constructed to violate the low-bond-dimension assumption; such controls would be required to delineate the boundary of applicability for the claimed polynomial scaling.
minor comments (2)
  1. [Introduction] Notation for the entanglement feature and its MPS bond dimension should be introduced with an explicit equation or definition in the main text rather than relying solely on the abstract.
  2. [Figures] Figure captions for the eigenstate applications should state the system sizes, Hamiltonian parameters, and TCI bond-dimension cutoff used in each panel to allow direct reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [Applications to eigenstates] Applications section: the identification of 'cases where eigenstates have entanglement feature learnable with TCI' reports only successful convergences and supplies neither reconstruction error bars, nor statistics on the fraction of eigenstates or system sizes for which TCI fails to converge, nor any scan of required bond dimension versus disorder or system size; this omission leaves the prevalence of the finite-bond-dimension property unquantified and weakens assessment of practical utility beyond the benchmark ensembles.

    Authors: We agree that the applications section would benefit from additional quantitative information to better assess practical utility. In the revised manuscript we will add reconstruction error bars for the reported eigenstate convergences, statistics on the fraction of eigenstates (and system sizes) for which TCI converges within the considered ensembles, and a limited scan of required bond dimension versus disorder strength for the disordered Hamiltonians studied. A exhaustive parameter scan across all system sizes remains computationally demanding, but the added data will help quantify prevalence where the finite-bond-dimension property holds. revision: yes

  2. Referee: [Benchmarks] Benchmark and method sections: while the TCI reconstruction matches analytic expectations on Haar-random and random-MPS states (where the MPS assumption holds by construction), the manuscript contains no explicit tests or failure-mode analysis on states deliberately constructed to violate the low-bond-dimension assumption; such controls would be required to delineate the boundary of applicability for the claimed polynomial scaling.

    Authors: We acknowledge that explicit failure-mode controls would strengthen the delineation of applicability. Although the polynomial scaling claim is explicitly conditional on the finite-bond-dimension assumption, we will add to the revised manuscript a concrete numerical example of a state whose entanglement feature has large bond dimension (for instance a product state expressed in a non-local basis), demonstrating that TCI requires super-polynomial samples when the assumption is violated. This will serve as an explicit boundary illustration. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is conditional on external algorithm and stated assumption

full rationale

The paper's central claim is explicitly conditional on the entanglement feature admitting a finite-bond-dimension MPS representation, which enables the polynomial scaling of the external TCI algorithm. TCI itself is an independently developed method (not derived or fitted within the paper). Benchmarks are performed on Haar-random and random-MPS states generated independently of the method, where the assumption holds by construction, and application to eigenstates is reported only where TCI converges. No load-bearing step reduces by construction to a self-definition, fitted input renamed as prediction, or self-citation chain. The result is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim depends on one domain assumption about the structure of the entanglement feature; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The entanglement feature is expressible as a finite bond dimension MPS
    This assumption is required for TCI to achieve polynomial sample complexity; it is stated explicitly in the abstract.

pith-pipeline@v0.9.0 · 5788 in / 1134 out tokens · 33163 ms · 2026-05-22T22:38:02.558364+00:00 · methodology

discussion (0)

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

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Solving the Gross-Pitaevskii equation on multiple different scales using the quantics tensor train representation

    quant-ph 2025-07 unverdicted novelty 5.0

    A quantics tensor train solver resolves the Gross-Pitaevskii equation across seven orders of magnitude in length scale in one dimension and on grids larger than a trillion points in two dimensions.

Reference graph

Works this paper leans on

71 extracted references · 71 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    EF of random Haar state Consider the EF of a random Haar state in the thermo- dynamic limit L → ∞, for a state of qudits. Intuitively, the expectation is that such states are volume law entan- gled, with a flat entanglement spectrum Sn ≃ (log d)LA where LA is the number of spins in the region A, where Sn is the n-th Renyi entropy. Naively, this would give...

    work page

  2. [2]

    magnetic field

    Random MPS of bond dimension φ Now we consider a random MPS state of bond dimen- sion φ with unnormalized density matrix ρ = |ψ ⟩ ⟨ψ|. The EF requires computing the purity of this state e−S2 = trρ2 A/(trρ)2, where the denominator is important to properly define the purity of unnormalized random state. To estimate this quantity, we use the tools devel- ope...

    work page

  3. [3]

    SYK model and partial tracing in fermionic systems The SYK GS state is the ground state of the SYK model [27]. The ensemble of Hamiltonians we consider is: HSYK = − √ 6 N3/2 X 1≤i<j<k<l≤L 4Jijkl χiχjχkχl (B1) where Jijkl are independent mean zero, variance one Gaussian random variables, and χi are Majorana fermions. We study the model numerically by group...

    work page

  4. [4]

    We start from a c-color Motzkin state

    Motzkin wave functions Now we briefly review additional area-law violating states. We start from a c-color Motzkin state. This state is defined on a space with local spin- c degrees of freedom. It is an equal-weight superposition of color- balanced Motzkin paths [30, 49]. Such path is a se- quence of steps directed up (1 , 1), down (1 , −1) or side- ways ...

    work page

  5. [5]

    They are also equal-weight superpositions, but of color- balanced Dyck paths, which are Motzkin paths without the sideway (1, 0) steps [28, 29, 49]

    analogues of c-color Motzkin states. They are also equal-weight superpositions, but of color- balanced Dyck paths, which are Motzkin paths without the sideway (1, 0) steps [28, 29, 49]. 16

    work page

  6. [6]

    + hL−1;L (B7) of two-site random local terms hj;k drawn from the Gaussian Unitary Ensemble (GUE)

    GUE-H states We also consider ground states of a random spin- 1 2 Hamiltonian constructed as a sum HGUE = h1;2 + h2;3 + . . . + hL−1;L (B7) of two-site random local terms hj;k drawn from the Gaussian Unitary Ensemble (GUE). These states model ground states of local Hamiltonians, and are labeled as GUE-H GS on the entanglement pattern map

    work page

  7. [7]

    Additional TCI complexity curves and raw entanglement feature distance data For completeness we add the entanglement feature complexity curves for the rest of the area-law violating states that are briefly discussed in the main text, see Fig. 9. We also clarify the intuition that the bond di- mension of the tensor-cross-interpolated EF closely fol- lows t...

    work page

  8. [8]

    Browaeys and T

    A. Browaeys and T. Lahaye, Many-body physics with individually controlled rydberg atoms, Nat. Phys.16, 132 (2020)

    work page 2020

  9. [9]

    Blatt and C

    R. Blatt and C. F. Roos, Quantum simulations with trapped ions, Nat. Phys. 8, 277 (2012)

    work page 2012

  10. [10]

    Gross and I

    C. Gross and I. Bloch, Quantum simulations with ultra- cold atoms in optical lattices, Science 357, 995 (2017)

    work page 2017

  11. [11]

    Monroe, W

    C. Monroe, W. C. Campbell, L.-M. Duan, Z.-X. Gong, A. V. Gorshkov, P. W. Hess, R. Islam, K. Kim, N. M. Linke, G. Pagano, P. Richerme, C. Senko, and N. Y. Yao, Programmable quantum simulations of spin systems with trapped ions, Rev. Mod. Phys. 93, 025001 (2021)

    work page 2021

  12. [12]

    Bloch, J

    I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008)

    work page 2008

  13. [13]

    Saffman, T

    M. Saffman, T. G. Walker, and K. Mølmer, Quantum information with rydberg atoms, Rev. Mod. Phys. 82, 2313 (2010)

    work page 2010

  14. [14]

    Polkovnikov, K

    A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalat- tore, Colloquium : Nonequilibrium dynamics of closed interacting quantum systems, Rev. Mod. Phys. 83, 863 (2011)

    work page 2011

  15. [15]

    W. S. Bakr, J. I. Gillen, A. Peng, S. F¨ olling, and M. Greiner, A quantum gas microscope for detecting sin- gle atoms in a hubbard-regime optical lattice, Nature 462, 74 (2009)

    work page 2009

  16. [16]

    Elben, S

    A. Elben, S. T. Flammia, H.-Y. Huang, R. Kueng, J. Preskill, B. Vermersch, and P. Zoller, The random- ized measurement toolbox, Nature Reviews Physics 5, 9 (2022)

    work page 2022

  17. [17]

    Vermersch, M

    B. Vermersch, M. Ljubotina, J. I. Cirac, P. Zoller, M. Ser- byn, and L. Piroli, Many-body entropies and entangle- ment from polynomially many local measurements, Phys. Rev. X 14, 031035 (2024)

    work page 2024

  18. [18]

    Rispoli, A

    M. Rispoli, A. Lukin, R. Schittko, S. Kim, M. E. Tai, J. L´ eonard, and M. Greiner, Quantum critical behaviour at the many-body localization transition, Nature 573, 385 (2019)

    work page 2019

  19. [19]

    A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, Quantum ther- malization through entanglement in an isolated many- body system, Science 353, 794 (2016)

    work page 2016

  20. [20]

    Schollw¨ ock, The density-matrix renormalization group, Rev

    U. Schollw¨ ock, The density-matrix renormalization group, Rev. Mod. Phys. 77, 259 (2005)

    work page 2005

  21. [21]

    D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Col- loquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019). 17

    work page 2019

  22. [22]

    Eisert, M

    J. Eisert, M. Cramer, and M. B. Plenio, Colloquium: Area laws for the entanglement entropy, Rev. Mod. Phys. 82, 277 (2010)

    work page 2010

  23. [23]

    Vidal ,\ 10.1103/PhysRevLett.93.040502 journal journal Phys

    G. Vidal, Efficient simulation of one-dimensional quan- tum many-body systems, Physical Review Letters 93, 10.1103/physrevlett.93.040502 (2004)

    work page doi:10.1103/physrevlett.93.040502 2004

  24. [24]

    A. A. Akhtar and Y.-Z. You, Multiregion entanglement in locally scrambled quantum dynamics, Physical Review B 102, 10.1103/physrevb.102.134203 (2020)

    work page doi:10.1103/physrevb.102.134203 2020

  25. [25]

    A. A. Akhtar, H.-Y. Hu, and Y.-Z. You, Scalable and flex- ible classical shadow tomography with tensor networks, Quantum 7, 1026 (2023)

    work page 2023

  26. [26]

    W.-T. Kuo, A. A. Akhtar, D. P. Arovas, and Y.-Z. You, Markovian entanglement dynamics under locally scrambled quantum evolution, Physical Review B 101, 10.1103/physrevb.101.224202 (2020)

    work page doi:10.1103/physrevb.101.224202 2020

  27. [27]

    Oseledets and E

    I. Oseledets and E. Tyrtyshnikov, TT-cross approxima- tion for multidimensional arrays, Linear Algebra and its Applications 432, 70 (2010)

    work page 2010

  28. [28]

    I. V. Oseledets, Tensor-train decomposition, SIAM Jour- nal on Scientific Computing 33, 2295 (2011)

    work page 2011

  29. [29]

    Savostyanov and I

    D. Savostyanov and I. Oseledets, Fast adaptive interpola- tion of multidimensional arrays in tensor train format, in The 2011 International Workshop on Multidimensional (nD) Systems (IEEE, 2011) pp. 1–8

    work page 2011

  30. [30]

    D. V. Savostyanov, Quasioptimality of maximum-volume cross interpolation of tensors, Linear Algebra and its Ap- plications 458, 217 (2014)

    work page 2014

  31. [31]

    Dolgov and D

    S. Dolgov and D. Savostyanov, Parallel cross interpola- tion for high-precision calculation of high-dimensional in- tegrals, Computer Physics Communications 246, 106869 (2020)

    work page 2020

  32. [32]

    Y. N. Fern´ andez, M. K. Ritter, M. Jeannin, J.-W. Li, T. Kloss, T. Louvet, S. Terasaki, O. Parcollet, J. von Delft, H. Shinaoka, and X. Waintal, Learning tensor net- works with tensor cross interpolation: new algorithms and libraries (2024), arXiv:2407.02454 [physics.comp-ph]

    work page arXiv 2024

  33. [34]

    Kitaev, A simple model of quantum holography, https://online.kitp.ucsb.edu/online/entangled15/ kitaev2/—https://online.kitp.ucsb.edu/online/ entangled15/kitaev2/ (2015)

    A. Kitaev, A simple model of quantum holography, https://online.kitp.ucsb.edu/online/entangled15/ kitaev2/—https://online.kitp.ucsb.edu/online/ entangled15/kitaev2/ (2015)

    work page 2015

  34. [35]

    Fredkin Spin Chain

    O. Salberger and V. Korepin, Fredkin spin chain (2016), arXiv:1605.03842 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  35. [37]

    Movassagh and P

    R. Movassagh and P. W. Shor, Supercritical entangle- ment in local systems: Counterexample to the area law for quantum matter, Proceedings of the National Academy of Sciences 113, 13278 (2016)

    work page 2016

  36. [38]

    Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326, 96 (2011)

    U. Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326, 96 (2011)

    work page 2011

  37. [39]

    H.-Y. Hu, S. Choi, and Y.-Z. You, Classical shadow tomography with locally scrambled quantum dynamics, Phys. Rev. Res. 5, 023027 (2023)

    work page 2023

  38. [40]

    Huang, R

    H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measure- ments, Nature Physics 16, 1050 (2020)

    work page 2020

  39. [41]

    Y.-Z. You, Z. Yang, and X.-L. Qi, Machine learning spa- tial geometry from entanglement features, Physical Re- view B 97, 10.1103/physrevb.97.045153 (2018)

    work page doi:10.1103/physrevb.97.045153 2018

  40. [43]

    Hayden, D

    P. Hayden, D. W. Leung, and A. Winter, Aspects of generic entanglement, Communications in Mathematical Physics 265, 95 (2006)

    work page 2006

  41. [44]

    Cirac, D

    I. Cirac, D. Perez-Garcia, N. Schuch, and F. Verstraete, Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems, arXiv e-prints , arXiv:2011.12127 (2020), arXiv:2011.12127 [quant-ph]

    work page arXiv 2011

  42. [45]

    Bianchi and P

    E. Bianchi and P. Don` a , Typical entanglement entropy in the presence of a center: Page curve and its variance, Physical Review D 100, 10.1103/physrevd.100.105010 (2019)

    work page doi:10.1103/physrevd.100.105010 2019

  43. [46]

    Haferkamp, C

    J. Haferkamp, C. Bertoni, I. Roth, and J. Eisert, Emer- gent statistical mechanics from properties of disordered random matrix product states, PRX Quantum 2, 040308 (2021)

    work page 2021

  44. [47]

    Coser, L

    A. Coser, L. Tagliacozzo, and E. Tonni, On r´ enyi en- tropies of disjoint intervals in conformal field theory, Journal of Statistical Mechanics: Theory and Experi- ment 2014, P01008 (2014)

    work page 2014

  45. [48]

    D. A. Roberts, D. Stanford, and L. Susskind, Local- ized shocks, Journal of High Energy Physics 2015, 10.1007/jhep03(2015)051 (2015)

    work page doi:10.1007/jhep03(2015)051 2015

  46. [50]

    D. J. Luitz, N. Laflorencie, and F. Alet, Many-body local- ization edge in the random-field Heisenberg chain, Phys. Rev. B 91, 081103 (2015)

    work page 2015

  47. [51]

    D. J. Luitz, Long tail distributions near the many-body localization transition, Phys. Rev. B 93, 134201 (2016)

    work page 2016

  48. [52]

    P. T. Dumitrescu, R. Vasseur, and A. C. Potter, Scaling theory of entanglement at the many-body localization transition, Phys. Rev. Lett. 119, 110604 (2017)

    work page 2017

  49. [53]

    Chen, Z.-C

    X. Chen, Z.-C. Gu, and X.-G. Wen, Local unitary trans- formation, long-range quantum entanglement, wave func- tion renormalization, and topological order, Phys. Rev. B 82, 155138 (2010)

    work page 2010

  50. [54]

    Horodecki, P

    R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Reviews of Mod- ern Physics 81, 865 (2009)

    work page 2009

  51. [55]

    Stress majorization implementation, https: //juliagraphs.org/NetworkLayout.jl/stable/ #Stress-Majorization, accessed: 2024-11-12

    work page 2024

  52. [56]

    Sugino and V

    F. Sugino and V. Korepin, R´ enyi entropy of highly entan- gled spin chains, International Journal of Modern Physics B 32, 1850306 (2018)

    work page 2018

  53. [57]

    Fishman, S

    M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor Software Library for Tensor Network Calcula- tions, SciPost Phys. Codebases , 4 (2022)

    work page 2022

  54. [58]

    Sozykin, A

    K. Sozykin, A. Chertkov, R. Schutski, A.-H. Phan, A. Ci- chocki, and I. Oseledets, TTOpt: a maximum volume quantized tensor train-based optimization and its appli- cation to reinforcement learning, in Proceedings of the 36th International Conference on Neural Information 18 Processing Systems, NIPS ’22 (Curran Associates Inc., Red Hook, NY, USA, 2022)

    work page 2022

  55. [59]

    S. A. Goreinov, I. V. Oseledets, D. V. Savostyanov, E. E. Tyrtyshnikov, and N. L. Zamarashkin, How to find a good submatrix, in Matrix Methods: Theory, Algorithms and Applications , pp. 247–256

    work page

  56. [60]

    Goremykina, R

    A. Goremykina, R. Vasseur, and M. Serbyn, Analytically solvable renormalization group for the many-body local- ization transition, Phys. Rev. Lett. 122, 040601 (2019)

    work page 2019

  57. [62]

    Provided, we allow basis ro- tations, the errors from interpolated elements can spread to all entries in the rotated basis

    TCI algorithm is guaranteed to have no errors in the en- tries of the EF that were used as pivots, while all remain- ing entries are interpolated. Provided, we allow basis ro- tations, the errors from interpolated elements can spread to all entries in the rotated basis. For the EF however, basis rotations are not a natural operation since it would mix tog...

    work page

  58. [63]

    Jeannin, Y

    M. Jeannin, Y. N´ u˜ nez Fern´ andez, T. Kloss, O. Parcol- let, and X. Waintal, Cross-extrapolation reconstruction of low-rank functions and application to quantum many- body observables in the strong coupling regime, Phys. Rev. B 110, 035124 (2024)

    work page 2024

  59. [64]

    N´ u˜ nez Fern´ andez, M

    Y. N´ u˜ nez Fern´ andez, M. Jeannin, P. T. Dumitrescu, T. Kloss, J. Kaye, O. Parcollet, and X. Waintal, Learning feynman diagrams with tensor trains, Phys. Rev. X 12, 041018 (2022)

    work page 2022

  60. [65]

    Ishida, N

    H. Ishida, N. Okada, S. Hoshino, and H. Shinaoka, Low- rank quantics tensor train representations of feynman di- agrams for multiorbital electron-phonon models (2024)

    work page 2024

  61. [66]

    Murray, H

    M. Murray, H. Shinaoka, and P. Werner, Nonequilibrium diagrammatic many-body simulations with quantics ten- sor trains, Physical Review B 109, 165135 (2024)

    work page 2024

  62. [67]

    M. K. Ritter, Y. N´ u˜ nez Fern´ andez, M. Wallerberger, J. von Delft, H. Shinaoka, and X. Waintal, Quantics tensor cross interpolation for high-resolution parsimo- nious representations of multivariate functions, Phys. Rev. Lett. 132, 056501 (2024)

    work page 2024

  63. [68]

    Shinaoka, M

    H. Shinaoka, M. Wallerberger, Y. Murakami, K. Nogaki, R. Sakurai, P. Werner, and A. Kauch, Multiscale space- time ansatz for correlation functions of quantum systems based on quantics tensor trains, Physical Review X 13, 021015 (2023)

    work page 2023

  64. [69]

    Takahashi, R

    H. Takahashi, R. Sakurai, and H. Shinaoka, Compact- ness of quantics tensor train representations of local imaginary-time propagators (2024)

    work page 2024

  65. [70]

    Sakaue, H

    K. Sakaue, H. Shinaoka, and R. Sakurai, Learning tensor trains from noisy functions with application to quantum simulation (2024)

    work page 2024

  66. [71]

    Hayden, S

    P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter, and Z. Yang, Holographic duality from random tensor networks, Journal of High Energy Physics2016, 9 (2016)

    work page 2016

  67. [72]

    Vasseur, A

    R. Vasseur, A. C. Potter, Y.-Z. You, and A. W. W. Lud- wig, Entanglement transitions from holographic random tensor networks, Phys. Rev. B 100, 134203 (2019)

    work page 2019

  68. [73]

    Note that in one dimension it is also possible to go around this normalization problem for random MPS [68, 69], but since we are after nonlinear properties like the typical en- tanglement spectrum of the EF, a replica trick is needed in any case

    work page

  69. [74]

    N. T. Vidal, M. L. Bera, A. Riera, M. Lewenstein, and M. N. Bera, Quantum operations in an information the- ory for fermions, Physical Review A 104, 10.1103/phys- reva.104.032411 (2021)

    work page doi:10.1103/phys- 2021

  70. [75]

    G. Lami, J. De Nardis, and X. Turkeshi, Anticoncentra- tion and state design of random tensor networks, Phys. Rev. Lett. 134, 010401 (2025)

    work page 2025

  71. [76]

    L´ oio, G

    H. L´ oio, G. Cecile, S. Gopalakrishnan, G. Lami, and J. D. Nardis, Correlations, spectra and entanglement transitions in ensembles of matrix product states (2025), arXiv:2412.14261 [quant-ph]

    work page arXiv 2025