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REVIEW 3 major objections 6 minor 88 references

Quantics Tensor Train for solving Gross-Pitaevskii equation

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read By keeping the Gross-Pitaevskii evolution inside a quantics tensor-train manifold, this paper claims storage and per-step cost grow only polylogarithmically with the number of spatial grid points, while matching standard solvers in…

desk verdict Plausible QTT-GPE solver with genuinely useful tensor-network constructions, but the central polylogarithmic-cost claim for real-time dynamics is not supported by the presented data. read the letter →

arxiv 2507.03134 v1 pith:YDCPMWSK submitted 2025-07-03 cond-mat.quant-gas

Aleix Bou-Comas , Marcin P{l}odzień , Luca Tagliacozzo , Juan José García-Ripoll This is my paper · ORCID
classification cond-mat.quant-gas
keywords Gross-PitaevskiiequationquanticstensortrainmatrixproductstatesnetworksnonlinearSchrödingerBose-Einsteincondensateimaginary-timeevolutionRunge-Kuttaintegrator
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that the one-dimensional Gross-Pitaevskii equation can be solved entirely in a quantics tensor-train (QTT) representation, a compressed form of the wavefunction on a binary-discretized grid. Both the nonlinear term $|\psi|^2\psi$ and long-range or multi-species interactions are implemented as tensor-network operations, so imaginary-time ground-state projection and real-time Runge-Kutta evolution never leave the low-rank manifold. If correct, storage and per-step cost grow only polylogarithmically with the number of grid points instead of linearly or quadratically, which would make very fine 1D simulations and large parameter sweeps practical on classical hardware. The paper supports the claim with ground-state and dynamics benchmarks for harmonic traps, quasi-disordered potentials, Rydberg-dressed long-range interactions, and two-component spinor condensates.

What carries the argument

The load-bearing object is the quantics tensor train (QTT), a matrix product state encoding of a function on a grid of $2^n$ points in which the coordinate is binary-discretized, so that the vector of function values becomes a rank-$n$ tensor with two indices per bit. Differential operators, translations, and convolutions are represented as matrix product operators built with the ripple-carry adder, and the nonlinear term $|\psi|^2\psi$ is constructed with a copy tensor that contracts $\psi$ and $\psi^*$ along matched physical indices, followed by a rank truncation that keeps the state inside the low-rank manifold. The copy tensor and its generalizations are what allow the method to stay in QTT form while adding nonlinearities, long-range potentials, and multiple species.

What would settle it

Run the real-time solver for a fixed target accuracy and measure the maximum bond dimension, the internal width of the compressed representation, as the grid is refined from $2^{12}$ to $2^{16}$ points and as propagation time grows; if it grows like a power of the grid size or linearly in time, the claimed polylogarithmic scaling is false.

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Extended reading notes

Core claim

The central claim is that the Gross-Pitaevskii equation is not inherently incompatible with tensor-network compression: by quantizing the spatial coordinate, so that a grid point's binary digits become the physical indices of a matrix product state, the wavefunction, the Hamiltonian, the local nonlinearity, the convolution kernel for long-range interactions, and even multiple species can all be expressed as QTT objects of small bond dimension. The nonlinear term is built with a copy tensor that forms $|\psi|^2$ from $\psi$ and $\psi^*$, and the resulting bond dimension $\chi^2$ is truncated back to a smaller $\chi'$ at each step; the authors report that $\chi'<\chi^2$ in their experiments. Two algorithms then run entirely inside this manifold: an imaginary-time projector that converges to the variational ground state, and a rank-adapted fourth-order Runge-Kutta integrator that advances real-time dynamics while actively truncating QTT ranks under a controlled error threshold. On their benchmarks the QTT results agree with standard solvers, while the measured maximum bond dimension grows only mildly, from $\chi=8$ at $2^9$ grid points to $\chi=9$ at $2^{12}$ points in the double-well example, which is what turns the exponential representation into polylogarithmic cost.

Load-bearing premise

The entire speedup rests on the unproven premise that the wavefunction and its self-interaction term, once written in the binary-grid tensor format, keep their internal complexity small as the grid is refined and as the condensate evolves; the paper demonstrates this for its examples but gives no general growth bound for that internal complexity.

Editorial extensions

If this is right

  • Ground-state and real-time simulations of one-dimensional condensates can be run with storage and per-step operation counts that scale like $O(\log N_{\rm grid})$ times a bond-dimension factor, in place of the $O(N_{\rm grid})$ or $O(N_{\rm grid}^2)$ costs of standard dense and dense-operator methods.
  • The same copy-tensor toolbox covers local nonlinearities, long-range convolution interactions, and an arbitrary number of BEC species, so the solver is not restricted to the simplest contact-interaction GPE.
  • For a fixed problem, once the grid is fine enough (around $2^{10}$ points in the paper's double-well example), the QTT storage cost falls below the cost of storing the dense wavefunction, and the gap widens as the grid is refined.
  • Because every step of both algorithms remains inside the low-rank manifold, the method can be used as a classical stand-in for quantum PDE solvers, giving a practical way to benchmark nonlinear Schrödinger dynamics without a quantum processor.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The real advantage over sparse-matrix methods is asymptotic: for the grid sizes reported, a sparse finite-difference matrix-vector product costs fewer operations than the QTT contraction, so the claimed exponential reduction only pays off when the required resolution is high enough to make sparse storage impractical.
  • If the mild bond-dimension growth observed in 1D carries over to 2D and 3D, a dimension-agnostic QTT solver would inherit polylogarithmic scaling in each coordinate; the paper explicitly leaves this as an extension.
  • A direct target for future work is long-time propagation of density patterns with developing fine structure (shocks, dark-soliton collisions, or strongly attractive collapse), since the unproven assumption is that the internal complexity of the representation stays bounded throughout the evolution, not just for stationary states and moderate times.
  • Replacing the copy-tensor multiplication with the faster TT multiplication algorithm acknowledged in footnote [63] would lower the leading per-step cost of the nonlinear term from $\chi^4$ to $\chi^3$, a concrete way to widen the regime where the method beats conventional solvers.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. This manuscript presents a QTT-based numerical method for the one-dimensional Gross-Pitaevskii equation. The authors construct QTT representations of the non-linear term |ψ|², local potentials, long-range convolution potentials, and multiple BEC species, and combine them with an imaginary-time projector for ground states and a rank-adapted RK4 integrator for real-time dynamics. They apply the method to harmonic-trap and double-well ground states, Aubry-André-Harper quasi-disorder dynamics, Rydberg-dressed long-range interactions with breathing oscillations, and two-component spinor dynamics. The paper claims that, because all operations stay within the low-rank QTT manifold, the computational cost grows only polylogarithmically with the number of grid points and that the method achieves an exponential reduction in resources while retaining quantitative accuracy.

Significance. The algorithmic toolbox is presented clearly and the physical scenarios are relevant. The construction of the multispecies QTT encoding with a generalized copy tensor, and the treatment of the nonlinear term with subsequent truncation, are useful additions to the QTT methods literature. The paper ships no code and no machine-checked proofs; its main value would be the numerical demonstration of a scalable classical alternative for nonlinear Schrödinger-type PDEs. If the polylogarithmic scaling and quantitative-accuracy claims were substantiated, this would be a valuable contribution. As it stands, however, the quantitative evidence covers only one static ground-state example, and the central dynamical claim remains an assumption rather than a demonstrated result.

major comments (3)
  1. [Abstract, §V.B, §VII] The central claim—that the full GPE evolution has cost growing only polylogarithmically with the grid size while retaining quantitative accuracy—is not supported by the reported data. The only concrete resource numbers are for a static ground state (χ=8 at 2⁹ grid points and χ=9 at 2¹² grid points, §VII). No bond-dimension time series, no rank distributions after the RK4 stages, and no truncation-error or convergence-versus-threshold data are reported for any real-time run. Section IV.A states that the raw nonlinear term has bond dimension χ² and must be truncated to χ′<χ², and Section V.B applies rank truncation after every RK4 stage, but no measured values of χ′ or its growth with time, grid size, or interaction strength are given. The phrase 'controlled error threshold' in Section VIII is therefore not backed by any threshold value or measured error. The advertised exponential/polylogarithmic advantage collapses to an empirical observation for one static case unless this evidence is supplied.
  2. [§VII, final paragraph] The claimed extension of the resource savings to higher dimensions is quantitatively incorrect as written. For a d-dimensional grid with N points per dimension, the QTT representation requires O(d log N χ²) parameters, not O(3N_grid) or O(MdN_grid) as stated in the text. The comparison with standard storage O(N^d) and O(N^{Md}) should be made explicitly with the logarithmic factor in place; as written, the claimed QTT advantage in d>1 is overstated by a factor that is exponential in d. This paragraph should be corrected because the polylogarithmic scaling is the paper's main quantitative selling point.
  3. [§VI.A, §VI.D] Statements such as 'full agreement up to numerical precision with standard GP solvers' and 'in full agreement with results presented in [89]' are not supported by any quantitative comparison. The paper reports no norms, difference plots, error bars, or solver parameters for these benchmarks. Since quantitative accuracy is one half of the central claim, the authors should report at least maximum absolute or L2 differences against a standard solver for each benchmark, together with the truncation tolerance and time step used.
minor comments (6)
  1. [Throughout] There are many typographical errors: 'Kroenecker' should be 'Kronecker' (several places, e.g. §IV.B), 'Ryderg' should be 'Rydberg' (§VI.C), 'Pitaesvkii' appears in §V.A, 'ACKNOLWEDGMENTS' in the acknowledgments, and 'the tern' in Eq. (7) should be 'the term'. A careful proofread is needed.
  2. [Fig. 6 caption] The caption states that 'dark green lines represent a BEC with repulsive interactions (g < 0)', but the text in §VI.B and the following sentence of the caption identify g < 0 with attractive interactions; this inconsistency should be fixed.
  3. [§V, §VI] The numerical parameters required for reproducibility—time step dt, convergence tolerance ϵ, rank-truncation tolerance, grid sizes, domain lengths, and numbers of imaginary-time steps—are not reported anywhere. Please add them, ideally in a table.
  4. [References] References [42] and [54] are the same work, as are [43] and [76]; these duplicates should be consolidated.
  5. [§III, around Eq. (13)] The sentence 'The convergence of the integral can be faster using different quadratures [55], but it is always exponential in the number of qubits' is too strong without qualification; for non-smooth functions the trapezoidal-rule convergence need not be exponential, and the statement should be reworded.
  6. [Fig. 10] The QTT operation counts in Fig. 10 are theoretical upper bounds, whereas the sparse-matrix counts are exact operation counts for the standard method; the caption should state this distinction so that the comparison is not read as measured runtimes.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the QTT-GPE derivation is self-contained and its resource claims are conditional benchmarks, not fits.

full rationale

The paper derives the GPE from the standard mean-field reduction (Section II), adapts standard QTT/MPO primitives (Sections III-IV), and then applies textbook imaginary-time and RK4 integrators (Section V). No parameter is fitted to the target outputs: the bond dimensions reported in Section VII are measured quantities, and the polylogarithmic cost statement in Section VIII is explicitly conditional on bond dimensions growing mildly, which the paper supports with numerical examples rather than deriving from its own assumptions. Several cited building blocks (e.g., Refs. [41]-[43], [76]) are from the same group, but they are used for implementation details (function loading, SVD truncation) and are not the load-bearing justification for the central compressibility claim, which rests on external TT/QTT results ([25]-[28], [49]-[52]) and on direct comparisons with standard GP solvers. The lack of bond-dimension growth data for the real-time RK4 solver is a completeness/evidence limitation, not circularity: the paper does not define the predicted cost in terms of a fitted value or infer the conclusion from a self-citation chain. Therefore the circularity score is 0.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities. It relies on standard QTT mathematical results, the physical validity of the GPE, and the numerical stability of the chosen integrators. The main uncontrolled element is the bond dimension growth during evolution, which is treated as an observed property rather than a proven bound.

free parameters (1)
  • Maximum bond dimension chi and rank-truncation tolerance = not reported
    The QTT solver's accuracy and cost depend on the chosen bond dimension and truncation tolerance; the paper reports chi values for one example but does not provide a systematic selection criterion.
assumptions (4)
  • standard math QTT compression bounds and finite-difference discretization provide accurate approximations of functions and operators (Section III).
    The method rests on known QTT rank bounds and on finite-difference translation MPOs; these are standard numerical analysis results cited from prior work.
  • domain assumption The Gross-Pitaevskii equation is a valid mean-field description of the dilute BECs considered, and the 1D reduction via tight transverse confinement is valid (Section II).
    All simulations use the 1D GPE, which is only an effective model derived under specific trapping assumptions.
  • domain assumption Imaginary-time evolution converges to the ground state when the initial state has nonzero overlap with the target state (Section V.A).
    This is a standard property of imaginary-time projectors, but the global convergence for the nonlinear GPE is not proven in the paper.
  • domain assumption The RK4 integrator with rank truncation remains stable and accurate for the GPE over the simulated time spans (Section V.B).
    No stability or error analysis is provided for the rank-adaptive RK4 scheme; the paper only reports qualitative agreement with expected dynamics.

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Cite this review

Pith. "Pith review of Quantics Tensor Train for solving Gross-Pitaevskii equation." pith.science (2026). https://pith.science/paper/YDCPMWSK

@misc{pith2026250703134,
  author       = {Pith},
  title        = {Pith review of: Quantics Tensor Train for solving Gross-Pitaevskii equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YDCPMWSK}},
  note         = {Machine review of arXiv:2507.03134}
}
read the original abstract

We present a quantum-inspired solver for the one-dimensional Gross-Pitaevskii equation in the Quantics Tensor-Train (QTT) representation. By evolving the system entirely within a low-rank tensor manifold, the method sidesteps the memory and runtime barriers that limit conventional finite-difference and spectral schemes. Two complementary algorithms are developed: an imaginary-time projector that drives the condensate toward its variational ground state and a rank-adapted fourth-order Runge-Kutta integrator for real-time dynamics. The framework captures a broad range of physical scenarios - including barrier-confined condensates, quasi-random potentials, long-range dipolar interactions, and multicomponent spinor dynamics - without leaving the compressed representation. Relative to standard discretizations, the QTT approach achieves an exponential reduction in computational resources while retaining quantitative accuracy, thereby extending the practicable regime of Gross-Pitaevskii simulations on classical hardware. These results position tensor networks as a practical bridge between high-performance classical computing and prospective quantum hardware for the numerical treatment of nonlinear Schrodinger-type partial differential equations.

Figures

Figures reproduced from arXiv: 2507.03134 by the authors.

Figure 1
Figure 1. FIG. 1. Three possible representations of how to store a [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Illustration of different formalism explained in sec [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Illustration of different tools to simulate GPE with different BEC species described in section [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Ground state of the GP equation with different inter [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Ground state of the GP equation with different interaction strength ( [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Ground state probability density of the GP equa [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Panels (a)-(i): color encoded density of the wave [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Panels (a)-(d) present color-encoded evolution of [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Storage (a) and computational (b) cost of storing [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. The panels (a) and (b) depict a positive and nega [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]

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Works this paper leans on

88 extracted references · 51 canonical work pages

  1. [89]

    J. E. Johnson and S. L. Rolston, Interactions be- tween rydberg-dressed atoms, Physical Review A 82, 10.1103/physreva.82.033412 (2010)

    work page doi:10.1103/physreva.82.033412 2010

  2. [1]

    European Union NextGenerationEU/PRTR

    From left to right, the first sum is the addition of (0 + 0 + 0) mod 2 = 0 , the second and the third is 16 (a) (c)(b) FIG. 11. The panels (a) and (b) depict a positive and nega- tive translation operator. In this casea = 1 4 + 1 16, the filled (empty) triangles mean thatai = 1 (ai = 0). Per convention, we choose that the triangles pointing down (up) will...

    arXiv

  3. [2]

    Montanaro and S

    A. Montanaro and S. Pallister, Quantum algorithms and the finite element method, Phys. Rev. A 93, 032324 (2016)

    2016

  4. [3]

    A. M. Childs, J.-P. Liu, and A. Ostrander, High-precision quantum algorithms for partial differential equations, Quantum 5, 574 (2021)

    2021

  5. [4]

    A. M. Childs and J.-P. Liu, Quantum spectral methods for differential equations, Communications in Mathemat- ical Physics375, 1427 (2020)

    2020

  6. [5]

    Kyriienko, A

    O. Kyriienko, A. E. Paine, and V. E. Elfving, Solving nonlinear differential equations with differentiable quan- tum circuits, Phys. Rev. A103, 052416 (2021)

    2021

  7. [6]

    A. E. Paine, V. E. Elfving, and O. Kyriienko, Quantum kernel methods for solving regression problems and dif- ferential equations, Phys. Rev. A107, 032428 (2023)

    2023

  8. [7]

    H.-Y. Wu, A. E. Paine, E. Philip, A. A. Gentile, and O. Kyriienko, Quantum algorithm for solving nonlin- ear differential equations based on physics–informed ef- fective hamiltonians, arXiv preprint arXiv:2504.13174 10.48550/arXiv.2504.13174 (2025), preprint

    arXiv doi:10.48550/arxiv.2504.13174 2025

Show all 88 references
  1. [8]

    Jin and N

    S. Jin and N. Liu, Quantum algorithms for nonlinear par- tial differential equations, Bulletin des Sciences Mathé- matiques 194, 103457 (2024)

    2024

  2. [9]

    Farghadan, M

    A. Farghadan, M. M. M. Farahani, and M. Akbari, A fast quantum algorithm for solving partial differential equa- tions, Scientific Reports15, 5317 (2025)

    2025

  3. [10]

    J. Hu, S. Jin, N. Liu, and L. Zhang, Quantum circuits for partial differential equations via schrödingerisation, Quantum 8, 1563 (2024)

    2024

  4. [11]

    S. Chen, J. Cotler, H.-Y. Huang, and J. Li, The complex- ity of NISQ, Nature Communications14, 6001 (2023)

    2023

  5. [12]

    Verstraete, V

    F. Verstraete, V. Murg, and J. I. Cirac, Matrix prod- uct states, projected entangled pair states, and varia- tional renormalization group methods for quantum spin systems, Advances in Physics57, 143 (2008)

    2008

  6. [13]

    J. I. Cirac and F. Verstraete, Renormalization and ten- sor product states in spin chains and lattices, Journal of Physics A: Mathematical and Theoretical42, 504004 (2009)

    2009

  7. [14]

    Eisert, M

    J. Eisert, M. Cramer, and M. B. Plenio, Area laws for the entanglement entropy, Reviews of Modern Physics 82, 277 (2010)

    2010

  8. [15]

    U.Schollwöck,Thedensity-matrixrenormalizationgroup in the age of matrix product states, Annals of Physics 326, 96 (2011)

    2011

  9. [16]

    Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Annals of Physics349, 117 (2014)

    R. Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Annals of Physics349, 117 (2014)

    2014

  10. [17]

    Biamonte and V

    J. Biamonte and V. Bergholm, Tensor networks in a nut- shell (2017), arXiv:1708.00006 [quant-ph]

    2017 arXiv

  11. [18]

    Orús, Tensor networks for complex quantum systems, Nature Reviews Physics1, 538 (2019)

    R. Orús, Tensor networks for complex quantum systems, Nature Reviews Physics1, 538 (2019)

    2019

  12. [19]

    Biamonte, Lectures on quantum tensor networks (2020), arXiv:1912.10049 [quant-ph]

    J. Biamonte, Lectures on quantum tensor networks (2020), arXiv:1912.10049 [quant-ph]

    2020 arXiv

  13. [20]

    S.-J. Ran, E. Tirrito, C. Peng, X. Chen, L. Tagliacozzo, G. Su, and M. Lewenstein,Tensor Network Contractions: Methods and Applications to Quantum Many-Body Sys- tems (Springer International Publishing, 2020)

    2020

  14. [21]

    Catarina and B

    G. Catarina and B. Murta, Density-matrix renormaliza- tion group: a pedagogical introduction, The European Physical Journal B96, 10.1140/epjb/s10051-023-00575- 2 (2023)

    2023 doi

  15. [22]

    Iblisdir, R

    S. Iblisdir, R. Orús, and J. I. Latorre, Matrix product states algorithms and continuous systems, Phys. Rev. B 75, 104305 (2007)

    2007

  16. [23]

    J. I. Latorre, Image compression and entanglement (2005), arXiv:quant-ph/0510031 [quant-ph]

    2005 arXiv

  17. [24]

    Evenbly and S

    G. Evenbly and S. R. White, Entanglement renormaliza- tion and wavelets, Phys. Rev. Lett.116, 140403 (2016)

    2016

  18. [25]

    Evenbly and S

    G. Evenbly and S. R. White, Representation and de- sign of wavelets using unitary circuits, Phys. Rev. A97, 052314 (2018)

    2018

  19. [26]

    I. V. Oseledets, Tensor-train decomposition, SIAM Jour- nal on Scientific Computing33, 2295 (2011)

    2011

  20. [27]

    V. A. Kazeev and B. N. Khoromskij, Low-rank explicit qtt representation of the laplace operator and its inverse, SIAM Journal on Matrix Analysis and Applications33, 742 (2012)

    2012

  21. [28]

    S. V. Dolgov, B. N. Khoromskij, and I. V. Os- eledets, Fast solution of parabolic problems in the ten- sor train/quantized tensor train format with initial ap- plication to the fokker–planck equation, SIAM Journal on Scientific Computing34, A3016 (2012)

    2012

  22. [29]

    Bigoni, A

    D. Bigoni, A. P. Engsig-Karup, and Y. M. Marzouk, Spectral tensor-train decomposition, SIAM Journal on Scientific Computing38, A2405 (2016)

    2016

  23. [30]

    J. Gidi, P. García-Molina, L. Tagliacozzo, and J. J. García-Ripoll, Pseudospectral method for solving pdes using matrix product states (2025), arXiv:2409.02916 [quant-ph]

    2025

  24. [31]

    A. M. P. Boelens, D. Venturi, and D. M. Tartakovsky, Parallel tensor methods for high-dimensional linear pdes, Journal of Computational Physics375, 519 (2018)

    2018

  25. [32]

    Dektor and D

    A. Dektor and D. Venturi, Dynamic tensor approxima- tion of high-dimensional nonlinear pdes, Journal of Com- putational Physics437, 110295 (2021)

    2021

  26. [33]

    Dektor, A

    A. Dektor, A. Rodgers, and D. Venturi, Rank-adaptive tensor methods for high-dimensional nonlinear pdes, Journal of Scientific Computing88, 36 (2021)

    2021

  27. [34]

    Richter, L

    L. Richter, L. Sallandt, and N. Nüsken, Solving high- dimensional parabolic pdes using the tensor train format, 18 in Proceedings of the 38th International Conference on Machine Learning(2021) pp. 8998–9009

    2021

  28. [35]

    D. P. Truong, M. I. Ortega, I. Boureima, G. Manzini, K. O. Rasmussen, and B. S. Alexandrov, Tensor networks for solving the time-independent boltzmann neutron transport equation, Journal of Computational Physics 507, 112943 (2024)

    2024

  29. [36]

    R. D. P. et al., Quantum-inspired framework for compu- tational fluid dynamics, Communications Physics7, 135 (2024)

    2024

  30. [37]

    Kiffner and D

    M. Kiffner and D. Jaksch, Tensor network reduced order models for wall-bounded flows, Physical Review Fluids 8, 124101 (2023)

    2023

  31. [38]

    Kornev, S

    E. Kornev, S. Dolgov, M. Perelshtein, and A. Melnikov, TetraFEM: Numerical Solution of Partial Differential Equations Using Tensor Train Finite Element Method, Mathematics 12, 3277 (2024)

    2024

  32. [39]

    N. G. et al., A quantum-inspired approach to exploit tur- bulence structures, Nature Computational Science2, 30 (2022)

    2022

  33. [40]

    X. L. et al., Functional tensor network approach to the many-body schrödinger equation, Physical Review B 105, 165116 (2022)

    2022

  34. [41]

    Ye and N

    E. Ye and N. F. G. Loureiro, Quantum-inspired method for solving the vlasov-poisson equations, Phys. Rev. E 106, 035208 (2022)

    2022

  35. [42]

    J. J. García-Ripoll, Quantum-inspired algorithms for multivariate analysis: from interpolation to partial dif- ferential equations, Quantum5, 431 (2021)

    2021

  36. [45]

    C. J. Pethick and H. Smith,Bose–Einstein Condensation in Dilute Gases(Cambridge University Press, 2008)

    2008

  37. [46]

    Affleck, T

    I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Rig- orous results on valence-bond ground states in antiferro- magnets, Phys. Rev. Lett.59, 799 (1987)

    1987

  38. [47]

    Fannes, B

    M. Fannes, B. Nachtergaele, and R. F. Werner, Exact antiferromagnetic ground states of quantum spin chains, Europhysics Letters10, 633 (1989)

    1989

  39. [48]

    S.ÖstlundandS.Rommer,Thermodynamiclimitofden- sity matrix renormalization, Phys. Rev. Lett.75, 3537 (1995)

    1995

  40. [49]

    Dukelsky, M

    J. Dukelsky, M. A. Martín-Delgado, T. Nishino, and G. Sierra, Equivalence of the variational matrix product method and the density matrix renormalization group applied to spin chains, Europhysics Letters 43, 457 (1998)

    1998

  41. [50]

    Oseledets and E

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

    2010

  42. [51]

    Lindsey, Multiscale interpolative construction of quantized tensor trains (2024), arXiv:2311.12554 [math.NA]

    M. Lindsey, Multiscale interpolative construction of quantized tensor trains (2024), arXiv:2311.12554 [math.NA]

    2024 arXiv

  43. [52]

    Ali and A

    M. Ali and A. Nouy, Approximation Theory of Tree Tensor Networks: Tensorized Univariate Functions – Part II, Constructive Approximation 58, 463 (2023), arXiv:2007.00128

    2023 arXiv

  44. [53]

    Ali and A

    M. Ali and A. Nouy, Approximation Theory of Tree Tensor Networks: Tensorized Univariate Functions, Con- structive Approximation58, 463 (2023)

    2023

  45. [54]

    Given that the computational cost increases with the bond dimension, we can estimate an upper bound of it by assuming that all the tensor share the largest bond dimension which we refer to asχ = maxi(χi)

  46. [55]

    J. J. Rodríguez-Aldavero, P. García-Molina, L. Taglia- cozzo, and J. J. García-Ripoll, Chebyshev approxima- tion and composition of functions in matrix product states for quantum-inspired numerical analysis (2025), arXiv:2407.09609 [quant-ph]

    2025 arXiv

  47. [56]

    Brass and K

    H. Brass and K. Petras,Quadrature theory: the theory of numerical integration on a compact interval, 178 (Amer- ican Mathematical Soc., 2011)

    2011

  48. [57]

    Núñez Fernández, M

    Y. Núñez Fernández, M. Jeannin, P. T. Dumitrescu, T. Kloss, J. Kaye, O. Parcollet, and X. Waintal, Learning Feynman Diagrams with Tensor Trains, Physical Review X 12, 041018 (2022)

    2022

  49. [58]

    M. K. Ritter, Y. Núñez Fernández, 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)

    2024

  50. [59]

    Tindall, M

    J. Tindall, M. Stoudenmire, and R. Levy, Compressing multivariate functions with tree tensor networks (2024), arXiv:2410.03572 [quant-ph]

    2024 arXiv

  51. [60]

    I. P. McCulloch, From density-matrix renormalization group to matrix product states, Journal of Statistical Me- chanics: Theory and Experiment2007, P10014 (2007)

    2007

  52. [61]

    J. Chen, E. Stoudenmire, and S. R. White, Quantum fourier transform has small entanglement, PRX Quan- tum 4, 040318 (2023)

    2023

  53. [62]

    Díez-Valle, F

    P. Díez-Valle, F. Martínez-García, J. J. García-Ripoll, and D. Porras, Learning Generalized Statistical Mechan- ics with Matrix Product States (2024), arXiv:2409.08352

    2024 arXiv

  54. [63]

    The cost is the same to the MPO-MPS contrac- tion O ( 4 log2(Ngrid)(χ3k +χ2k2) ) but in this case χ = k, therefore we arribe at a computational cost O ( 8 log2(Ngrid)χ4)

  55. [64]

    During the preparation of this manuscript a novel TT multiplication algorithm has been proposed in [90]. It reduces the computational complexity of such multipli- cations fromO(χ4) toO(χ3) and the memory cost from O(χ3)toO(χ2).Theuseofsuchalgorithmswouldfurther reduce the comp...

  56. [65]

    W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery,Numerical Recipes: The Art of Scientific Com- puting, 3rd ed. (Cambridge University Press, Cambridge, UK, 2007)

    2007

  57. [66]

    Antoine and R

    X. Antoine and R. Duboscq, GPELab, a Matlab toolbox to solve Gross–Pitaevskii equations I: Computation of stationary solutions, Computer Physics Communications 185, 2969 (2014)

    2014

  58. [67]

    Antoine and R

    X. Antoine and R. Duboscq, GPELab, a Matlab tool- box to solve Gross–Pitaevskii equations II: Dynamics and stochastic simulations, Computer Physics Communica- tions 193, 95 (2015)

    2015

  59. [68]

    G. R. Dennis, J. J. Hope, and M. T. Johnsson, XMDS2: Fast, scalable simulation of coupled stochastic partial dif- ferential equations, arXiv:1204.4255 (2012)

    2012 arXiv

  60. [69]

    J. R. Schloss and L. J. O’Riordan, GPUE: Graphics Pro- cessing Unit Gross–Pitaevskii Equation solver, Journal 19 of Open Source Software3, 1037 (2018)

    2018

  61. [70]

    R. M. Caplan, NLSEmagic: Nonlinear Schrödinger Equation Multidimensional Matlab-based GPU- accelerated Integrators using Compact High-order Schemes, arXiv:1203.1263 (2012)

    2012 arXiv

  62. [71]

    Gaidamour, Q

    J. Gaidamour, Q. Tang, and X. Antoine, BEC2HPC: a HPC spectral solver for nonlinear Schrödinger and Gross–Pitaevskii equations. Stationary states computa- tion, Computer Physics Communications 265, 108007 (2021)

    2021

  63. [72]

    Fioroni et al., A Python GPU-accelerated solver for the Gross–Pitaevskii equation and applications to many- body cavity QED, arXiv:2404.14401 (2024)

    L. Fioroni et al., A Python GPU-accelerated solver for the Gross–Pitaevskii equation and applications to many- body cavity QED, arXiv:2404.14401 (2024)

    2024 arXiv

  64. [73]

    Muruganandam and S

    P. Muruganandam and S. K. Adhikari, Fortran programs for the time-dependent Gross–Pitaevskii equation in a fully anisotropic trap, Computer Physics Communica- tions 180, 1888 (2009)

    2009

  65. [74]

    Vudragović, I

    D. Vudragović, I. Vidanović, A. Balaž, P. Muruganan- dam, and S. K. Adhikari, C programs for solving the time-dependent Gross–Pitaevskii equation in a fully anisotropic trap, Computer Physics Communications 183, 2021 (2012)

    2012

  66. [75]

    M. B. Soley, P. Bergold, A. A. Gorodetsky, and V. S. Batista, Functional tensor-train chebyshev method for multidimensional quantum dynamics simulations, Jour- nal of Chemical Theory and Computation18, 25 (2022), pMID: 34898201

    2022

  67. [76]

    Gourianov, M

    N. Gourianov, M. Lubasch, S. Dolgov, Q. Y. van den Berg, H. Babaee, P. Givi, M. Kiffner, and D. Jaksch, A quantum-inspired approach to exploit turbulence struc- tures, Nature Computational Science2, 30 (2022)

    2022

  68. [77]

    P.García-Molina, L.Tagliacozzo,andJ.J.García-Ripoll, Global optimization of mps in quantum-inspired numer- ical analysis (2024), arXiv:2303.09430 [quant-ph]

    2024

  69. [78]

    José García-Ripoll, Time evolution of Matrix Product States, New Journal of Physics8, 305 (2006)

    J. José García-Ripoll, Time evolution of Matrix Product States, New Journal of Physics8, 305 (2006)

    2006

  70. [80]

    P. G. Harper, Single Band Motion of Conduction Elec- trons in a Uniform Magnetic Field, Proceedings of the Physical Society. Section A68, 874 (1955)

    1955

  71. [81]

    Aubry and G

    S. Aubry and G. André, Analyticity breaking and Ander- son localization in incommensurate lattices, Proceedings, VIII International Colloquium on Group-Theoretical Methods in Physics3 (1980)

    1980

  72. [82]

    Dziurawiec, J

    M. Dziurawiec, J. O. de Almeida, M. L. Bera, M. Płodzień, M. M. Maśka, M. Lewenstein, T. Grass, and U. Bhattacharya, Unraveling multifractality and mobility edges in quasiperiodic aubry-andré-harper chains through high-harmonic generation (2023), arXiv:2310.02757 [cond-mat.dis-nn]

    2023 arXiv

  73. [83]

    Roati, C

    G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. In- guscio, Anderson localization of a non-interacting Bose– Einstein condensate, Nature453, 895 (2008)

    2008

  74. [84]

    Modugno, Exponential localization in one- dimensional quasi-periodic optical lattices, New Journal of Physics11, 033023 (2009)

    M. Modugno, Exponential localization in one- dimensional quasi-periodic optical lattices, New Journal of Physics11, 033023 (2009)

    2009

  75. [85]

    Deissler, M

    B. Deissler, M. Zaccanti, G. Roati, C. D’Errico, M. Fat- tori, M. Modugno, G. Modugno, and M. Inguscio, De- localization of a disordered bosonic system by repulsive interactions, Nature Physics6, 354–358 (2010)

    2010

  76. [86]

    Michal, B

    V. Michal, B. Altshuler, and G. Shlyapnikov, Delocaliza- tion of weakly interacting bosons in a 1d quasiperiodic potential, Physical Review Letters 113, 10.1103/phys- revlett.113.045304 (2014)

    2014 doi

  77. [87]

    S. Ray, M. Pandey, A. Ghosh, and S. Sinha, Localization of weakly interacting bose gas in quasiperiodic potential, New Journal of Physics18, 013013 (2015)

    2015

  78. [88]

    Płodzień, G

    M. Płodzień, G. Lochead, J. de Hond, N. J. van Druten, and S. Kokkelmans, Rydberg dressing of a one- dimensional bose-einstein condensate, Phys. Rev. A95, 043606 (2017)

    2017

  79. [90]

    T. Wang, Optimal point-wise error estimate of a compact difference scheme for the coupled gross–pitaevskii equa- tions in one dimension, Journal of Scientific Computing 59, 158–186 (2013)

    2013

  80. [91]

    A.A.Michailidis, C.Fenton,andM.Kiffner,Tensortrain multiplication (2024), arXiv:2410.19747 [physics.comp- ph]

    2024 arXiv

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