- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

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

Marcel Niedermeier1,2, Adrien Moulinas2, Thibaud Louvet2, Jose L. Lado1, and Xavier Waintal2,*

  • *Contact author: xavier.waintal@cea.fr
PDF

Phys. Rev. Research 8, 023006 – Published 2 April, 2026

DOI: https://doi.org/10.1103/zwn1-f981

Abstract

Solving partial differential equations across multiple length scales represents a formidable challenge where reaching high precision can require a prohibitive amount of computer memory or computing time. However, the solutions to physics problems typically have structures operating on different length scales, and as a result exhibit a high degree of compressibility. Here, we use the quantics tensor train representation to build a tensor network solver for the time-dependent Gross-Pitaevskii equation. We demonstrate that the quantics approach generalizes well to the presence of the nonlinear term in the equation. We show that we can resolve phenomena across length scales separated by seven orders of magnitude in one dimension within one hour on a single core in a laptop, greatly surpassing the capabilities of more naive methods. We illustrate our methodology with various modulated optical trap potentials presenting features at vastly different length scales, including solutions to the Gross-Pitaevskii equation on two-dimensional grids above a trillion points (220×220). This quantum-inspired methodology can be readily extended to other partial differential equations combining spatial and temporal evolutions, providing a powerful method to solve highly featured differential equations at unprecedented length scales.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (108)

  1. J. M. Arrazola, A. Delgado, B. R. Bardhan, and S. Lloyd, Quantum-inspired algorithms in practice, Quantum 4, 307 (2020).
  2. L. Chakhmakhchyan, N. J. Cerf, and R. Garcia-Patron, Quantum-inspired algorithm for estimating the permanent of positive semidefinite matrices, Phys. Rev. A 96, 022329 (2017).
  3. K.-H. Han and J.-H. Kim, Quantum-inspired evolutionary algorithm for a class of combinatorial optimization, IEEE Trans. Evol. Comput. 6, 580 (2002).
  4. A. Narayanan and M. Moore, Quantum-inspired genetic algorithms, in Proceedings of IEEE International Conference on Evolutionary Computation, ICEC-96 (IEEE, Nagoya, 1996).
  5. E. Tang, A  quantum-inspired classical algorithm for recommendation systems , in Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC’19 (ACM, New York, NY, 2019).
  6. T. F. Rønnow, Z. Wang, J. Job, S. Boixo, S. V. Isakov, D. Wecker, J. M. Martinis, D. A. Lidar, and M. Troyer, Defining and detecting quantum speedup, Science 345, 420 (2014).
  7. Y. Zhou, E. M. Stoudenmire, and X. Waintal, What limits the simulation of quantum computers? Phys. Rev. X 10, 041038 (2020).
  8. J. Tindall, M. Fishman, E. M. Stoudenmire, and D. Sels, Efficient tensor network simulation of IBM’s Eagle kicked Ising experiment, PRX Quantum 5, 010308 (2024).
  9. F. Pan, K. Chen, and P. Zhang, Solving the sampling problem of the Sycamore quantum circuits, Phys. Rev. Lett. 129, 090502 (2022).
  10. F. Pan and P. Zhang, Simulation of quantum circuits using the big-batch tensor network method, Phys. Rev. Lett. 128, 030501 (2022).
  11. J. Chen, E. Stoudenmire, and S. R. White, Quantum Fourier transform has small entanglement, PRX Quantum 4, 040318 (2023).
  12. S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992).
  13. U. Schollwöck, The density-matrix renormalization group, Rev. Mod. Phys. 77, 259 (2005).
  14. R. Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. 349, 117 (2014).
  15. H.-C. Jiang and T. P. Devereaux, Superconductivity in the doped Hubbard model and its interplay with next-nearest hopping t’ Science 365, 1424 (2019).
  16. H. Xu, C.-M. Chung, M. Qin, U. Schollwöck, S. R. White, and S. Zhang, Coexistence of superconductivity with partially filled stripes in the Hubbard model, Science 384, 10.1126/science.adh7691, 202 (2024).
  17. A. Szasz, J. Motruk, M. P. Zaletel, and J. E. Moore, Chiral spin liquid phase of the triangular lattice Hubbard model: A density matrix renormalization group study, Phys. Rev. X 10, 021042 (2020).
  18. S. Yan, D. A. Huse, and S. R. White, Spin-liquid ground state of the S = 1/2 kagome Heisenberg antiferromagnet, Science 332, 1173 (2011).
  19. B. Zanger, C. B. Mendl, M. Schulz, and M. Schreiber, Quantum algorithms for solving ordinary differential equations via classical integration methods, Quantum 5, 502 (2021).
  20. A. D. Fokker, Die mittlere energie rotierender elektrischer dipole im strahlungsfeld, Ann. Phys. 348, 810 (1914).
  21. M. Planck, Über einen Satz der statistischen Dynamik und seine Erweiterung in der Quantentheorie, Sitzungsberichte der K. Preuss. Akad. der Wiss. Phys.-math. Kl (Reimer, Berlin, 1917).
  22. R. F. Pawula, Approximation of the linear Boltzmann equation by the Fokker-Planck equation, Phys. Rev. 162, 186 (1967).
  23. R. Jordan, D. Kinderlehrer, and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal. 29, 1 (1998).
  24. E. P. Gross, Structure of a quantized vortex in  Boson systems, Il Nuovo Cimento 20, 454 (1961).
  25. L. P. Pitaevskii, Vortex lines in an imperfect bose gas, J. Exptl. Theoret. Phys. 40, 646 (1961).
  26. E. Runge and E. K. U. Gross, Density-functional theory for time-dependent systems, Phys. Rev. Lett. 52, 997 (1984).
  27. R. van Leeuwen, Mapping from densities to potentials in time-dependent density-functional theory, Phys. Rev. Lett. 82, 3863 (1999).
  28. K. Yee, Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Trans. Antennas Propag. 14, 302 (1966).
  29. P. Monk, Finite Element Methods for Maxwell’s Equations (Oxford University Press, Oxford, 2003).
  30. F. L. Teixeira, C. Sarris, Y. Zhang, D.-Y. Na, J.-P. Berenger, Y. Su, M. Okoniewski, W. C. Chew, V. Backman, and J. J. Simpson, Finite-difference time-domain methods, Nat. Rev. Methods Primers 3, 75 (2023).
  31. F. H. Harlow and J. E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids 8, 2182 (1965).
  32. R. Issa, Solution of the implicitly discretised fluid flow equations by operator-splitting, J. Comput. Phys. 62, 40 (1986).
  33. J. H. Ferziger and M. Perić, Computational Methods for Fluid Dynamics (Springer, Berlin, 2002).
  34. L. Landau, E. Lifshitz, et al., On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowjetunion 8, 101 (1935).
  35. T. Gilbert, Classics in magnetics a phenomenological theory of damping in ferromagnetic materials, IEEE Trans. Magn. 40, 3443 (2004).
  36. J. Fidler and T. Schrefl, Micromagnetic modelling—The current state of the art, J. Phys. D: Appl. Phys. 33, R135 (2000).
  37. R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems (Society for Industrial and Applied Mathematics, Philadelphia, 2007).
  38. J. Crank and P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, Math. Proc. Cambridge Philos. Soc. 43, 50 (1947).
  39. M. Feit, J. Fleck, and A. Steiger, Solution of the Schrödinger equation by a spectral method, J. Comput. Phys. 47, 412 (1982).
  40. C. Runge, Ueber die numerische Auflösung von differentialGleichungen, Math. Ann. 46, 167 (1895).
  41. N. Hatano and M. Suzuki, Finding exponential product formulas of higher orders, Quantum Annealing and Other Optimization Methods (Springer, Berlin, 2005), pp. 37–68.
  42. Y. Núñez Fernández, 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).
  43. A. O. Fumega, M. Niedermeier, and J. L. Lado, Correlated states in super-moiré materials with a kernel polynomial quantics tensor cross interpolation algorithm, 2D Mater. 12, 015018 (2025).
  44. M. Jeannin, Y. Núñez Fernández, T. Kloss, O. Parcollet, 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).
  45. N. Jolly, Y. Núñez Fernández, and X. Waintal, Tensorized orbitals for computational chemistry, Phys. Rev. B 111, 245115 (2025).
  46. M. K. Ritter, Y. Núñez Fernández, M. Wallerberger, J. von Delft, H. Shinaoka, and X. Waintal, Quantics tensor cross interpolation for high-resolution parsimonious representations of multivariate functions, Phys. Rev. Lett. 132, 056501 (2024).
  47. K. Sakaue, H. Shinaoka, and R. Sakurai, Adaptive sampling-based optimization of quantics tensor trains for noisy functions: applications to quantum simulations, Sci. Post. Phys. 19 038 (2024).
  48. H. Ishida, N. Okada, S. Hoshino, and H. Shinaoka, Low-rank quantics tensor train representations of Feynman diagrams for multiorbital electron-phonon models, Phys. Rev. Lett. 135, 046502 (2025).
  49. H. Takahashi, R. Sakurai, and H. Shinaoka, Compactness of quantics tensor train representations of local imaginary-time propagators, SciPost Phys. 18, 007 (2024).
  50. Y. Sun, M. Niedermeier, T. V. C. Antão, A. O. Fumega, and J. L. Lado, Self-consistent tensor network method for correlated super-moiré matter beyond one billion sites, Phys. Rev. Res. 7, 043288 (2025).
  51. T. V. C. Antão, Y. Sun, A. O. Fumega, and J. L. Lado, Tensor network method for real-space topology in quasicrystal Chern mosaics, arXiv:2506.05230.
  52. A. Moustaj, Y. Sun, T. V. C. Antão, and J. L. Lado, Momentum-resolved spectral functions of super-moiré systems using tensor networks, arXiv:2512.18397.
  53. M. Murray, H. Shinaoka, and P. Werner, Nonequilibrium diagrammatic many-body simulations with quantics tensor trains, Phys. Rev. B 109, 165135 (2024).
  54. 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, Phys. Rev. X 13, 021015 (2023).
  55. P. Haubenwallner and M. Heller, Fully numerical Hartree-Fock calculations with quantized tensor trains, Electron. Struct. 7, 025006 (2025).
  56. M. Maksymilian, K. Inayoshi, H. Shinaoka, and P. Werner, High-resolution nonequilibrium GW calculations based on quantics tensor trains, Phys. Rev. Lett. 135, 226501 (2025).
  57. A. J. Kim and P. Werner, Strong coupling impurity solver based on quantics tensor cross interpolation, Phys. Rev. B 111, 125120 (2025).
  58. S. Rohshap, M. K. Ritter, H. Shinaoka, J. von Delft, M. Wallerberger, and A. Kauch, Two-particle calculations with quantics tensor trains—Solving the parquet equations, Phys. Rev. Res. 7, 023087 (2025).
  59. L. Hölscher, P. Rao, L. Müller, J. Klepsch, A. Luckow, T. Stollenwerk, and F. K. Wilhelm, Quantum-inspired fluid simulation of two-dimensional turbulence with GPU acceleration, Phys. Rev. Res. 7, 013112 (2025).
  60. S.-X. Zhang and L. Wang, A unified variational framework for quantum excited states, arXiv:2504.21459.
  61. M. Usvyatsov, R. Ballester-Rippoll, L. Bashaeva, K. Schindler, G. Ferrer, and I. Oseledets, T4DT: Tensorizing time for learning temporal 3D visual data, arXiv:2208.01421.
  62. M. B. Soley, P. Bergold, and V. S. Batista, Iterative power algorithm for global optimization with quantics tensor trains, J. Chem. Theory Comput. 17, 3280 (2021).
  63. D. Kolisnyk, R. A. Medina, R. Vasseur, and M. Serbyn, Tensor cross interpolation of purities in quantum many-body systems, arXiv:2503.17230.
  64. M. Jeannin, Y. Núñez-Fernández, T. Kloss, O. Parcollet, and X. Waintal, A comprehensive study of out-of-equilibrium Kondo effect and Coulomb blockade, Phys. Rev. B 112, 155159 (2025).
  65. S. Dolgov and D. Savostyanov, Tensor cross interpolation for global discrete optimization with application to Bayesian network inference, Comput. Math. and Math. Phys. 65, 1591 (2025).
  66. S. B. Kožić and G. Torre, Computing quantum resources using tensor cross interpolation, arXiv:2502.06956.
  67. V. Bohun, I. Lukin, M. Luhanko, G. Korpas, P. J. S. De Brouwer, M. Maksymenko, and M. Koch-Janusz, Entanglement scaling in matrix product state representation of smooth functions and their shallow quantum circuit approximations, arXiv:2412.05202.
  68. M. Eckstein, Solving quantum impurity models in the non-equilibrium steady state with tensor trains, arXiv:2410.19707.
  69. J. José Rodríguez-Aldavero, P. García-Molina, L. Tagliacozzo, and J. José García-Ripoll, Chebyshev approximation and composition of functions in matrix product states for quantum-inspired numerical analysis, arXiv:2407.09609.
  70. A. Chertkov and I. Oseledets, Solution of the Fokker-Planck equation by cross approximation method in the tensor train format, Front. Artif. Intell. 4, 668215 (2021).
  71. L. Arenstein, M. Mikkelsen, and M. Kastoryano, Fast and flexible quantum-inspired differential equation solvers with data integration, arXiv:2505.17046.
  72. 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 structures, Nat. Comput. Sci. 2, 30 (2022).
  73. M. Lubasch, P. Moinier, and D. Jaksch, Multigrid renormalization, J. Comput. Phys. 372, 587 (2018).
  74. I. V. Oseledets, Approximation of matrices with logarithmic number of parameters, Dokl. Math. 80, 653 (2009).
  75. I. Oseledets and E. Tyrtyshnikov, Tt-cross approximation for multidimensional arrays, Linear Algebra Appl. 432, 70 (2010).
  76. I. V. Oseledets, Tensor-train decomposition, SIAM J. Sci. Comput. 33, 2295 (2011).
  77. Y. Núñez Fernández, 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 networks with tensor cross interpolation: New algorithms and libraries, SciPost Phys. 18, 104 (2025).
  78. S. Dolgov, B. Khoromskij, and D. Savostyanov, Superfast Fourier transform using QTT approximation, J. Fourier Anal. Appl. 18, 915 (2012).
  79. S. Dolgov and D. Savostyanov, Parallel cross interpolation for high-precision calculation of high-dimensional integrals, Comput. Phys. Commun. 246, 106869 (2020).
  80. B. N. Khoromskij, Tensor numerical methods for high-dimensional PDEs: Basic theory and initial applications, arXiv:1408.4053.
  81. T. Tsednee, B. Tsednee, and T. Khinayat, Numerical solution to the time-dependent Gross-Pitaevskii equation, J. Appl. Sci. Eng. A 3, 17 (2022).
  82. H. Wang, Numerical studies on the split-step finite difference method for nonlinear Schrödinger equations, Appl. Math. Comput. 170, 17 (2005).
  83. W. Bao, D. Jaksch, and P. A. Markowich, Numerical solution of the Gross–Pitaevskii equation for Bose–Einstein condensation, J. Comput. Phys. 187, 318 (2003).
  84. P. Muruganandam and S. Adhikari, Fortran programs for the time-dependent Gross–Pitaevskii equation in a fully anisotropic trap, Comput. Phys. Commun. 180, 1888 (2009).
  85. H. Wang, X. Ma, J. Lu, and W. Gao, An efficient time-splitting compact finite difference method for Gross–Pitaevskii equation, Appl. Math. Comput. 297, 131 (2017).
  86. M. L. Cerimele, M. L. M. Chiofalo, F. Pistella, S. Succi, and M. P. Tosi, Numerical solution of the Gross-Pitaevskii equation using an explicit finite-difference scheme: An application to trapped Bose-Einstein condensates, Phys. Rev. E 62, 1382 (2000).
  87. L. Fioroni, L. Gravina, J. Stefaniak, A. Baumgärtner, F. Finger, D. Dreon, and T. Donner, A Python GPU-accelerated solver for the Gross-Pitaevskii equation and applications to many-body cavity QED, SciPost Phys. Codebases 38 (2024).
  88. A. Bou-Comas, M. Płodzień, L. Tagliacozzo, and J.-J. García-Ripoll, Quantics tensor train for solving Gross-Pitaevskii equation, arXiv:2507.03134.
  89. R. J. J. Connor, C. W. Duncan, and A. J. Daley, Tensor network methods for the Gross-Pitaevskii equation on fine grids, New J. Phys. 28, 023203 (2026).
  90. Q.-C. Chen, I.-K. Liu, J.-W. Li, and C.-M. Chung, Solving the Gross-Pitaevskii equation with quantic tensor trains: Ground states and nonlinear dynamics, arXiv:2507.04279.
  91. X. Waintal, C.-H. Huang, and C. W. Groth, Who can compete with quantum computers? Lecture notes on quantum inspired tensor networks computational techniques, arXiv:2601.03035 [quant-ph].
  92. T. Ayral, T. Louvet, Y. Zhou, C. Lambert, E. M. Stoudenmire, and X. Waintal, Density-matrix renormalization group algorithm for simulating quantum circuits with a finite fidelity, PRX Quantum 4, 020304 (2023).
  93. U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. 326, 96 (2011).
  94. A. J. Leggett, Bose-einstein condensation in the alkali gases: Some fundamental concepts, Rev. Mod. Phys. 73, 307 (2001).
  95. S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G. V. Shlyapnikov, and M. Lewenstein, Dark solitons in Bose-Einstein condensates, Phys. Rev. Lett. 83, 5198 (1999).
  96. J. Chen and M. Lindsey, Direct interpolative construction of the discrete Fourier transform as a matrix product operator, Appl. Comput. Harmon. Anal. 81, 101817 (2026).
  97. S. Paeckel, T. Köhler, A. Swoboda, S. R. Manmana, U. Schollwöck, and C. Hubig, Time-evolution methods for matrix-product states, Ann. Phys. 411, 167998 (2019).
  98. M. Niedermeier (2025), https://github.com/marcelniedermeier/gross-pitaevskii-tci.
  99. M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor software library for tensor network calculations, SciPost Phys. Codebases 4 (2022).
  100. M. Ritter and contributors, Quanticstci.jl, (2022), email: Ritter.Marc@physik.uni-muenchen.de, https://github.com/tensor4all/quanticstci.jl.
  101. M. Ritter and contributors, Tensorcrossinterpolation.jl, (2022), email: Ritter.Marc@physik.uni-muenchen.de, https://github.com/tensor4all/TensorCrossInterpolation.jl/tree/main.
  102. J. R. Anglin and W. Ketterle, Bose–Einstein condensation of atomic gases, Nature (London) 416, 211 (2002).
  103. M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms, Nature (London) 415, 39 (2002).
  104. M. Schreiber, S. S. Hodgman, P. Bordia, H. P. Lüschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, Observation of many-body localization of interacting fermions in a quasirandom optical lattice, Science 349, 842 (2015).
  105. H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, Cold atoms in cavity-generated dynamical optical potentials, Rev. Mod. Phys. 85, 553 (2013).
  106. J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken, and F. Verstraete, Unifying time evolution and optimization with matrix product states, Phys. Rev. B 94, 165116 (2016).
  107. K. Viebahn, M. Sbroscia, E. Carter, J.-C. Yu, and U. Schneider, Matter-wave diffraction from a quasicrystalline optical lattice, Phys. Rev. Lett. 122, 110404 (2019).
  108. J.-C. Yu, S. Bhave, L. Reeve, B. Song, and U. Schneider, Observing the two-dimensional Bose glass in an optical quasicrystal, Nature (London) 633, 338 (2024).

Outline

Information

Phys. Rev. Research 8, 023006– Published 2 April, 2026
  • Received 9 July 2025
  • Accepted 13 February 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation