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Solving the Gross-Pitaevskii Equation with Quantic Tensor Trains: Ground States and Nonlinear Dynamics
by Qian-Can Chen, I-Kang Liu, Jheng-Wei Li, Chia-Min Chung
Submission summary
| Authors (as registered SciPost users): | Chia-Min Chung |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2507.04279v3 (pdf) |
| Code repository: | https://github.com/Heliumky/TN_GPE |
| Date submitted: | Dec. 11, 2025, 7:54 a.m. |
| Submitted by: | Chia-Min Chung |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We develop a tensor network framework based on the quantic tensor train (QTT) format to efficiently solve the Gross-Pitaevskii equation (GPE), which governs Bose-Einstein condensates under mean-field theory. By adapting time-dependent variational principle (TDVP) and gradient descent methods, we accurately handle the GPE's nonlinearities within the QTT structure. Our approach enables high-resolution simulations with drastically reduced computational cost. We benchmark ground states and dynamics of BECs--including vortex lattice formation and breathing modes--demonstrating superior performance over conventional grid-based methods and stable long-time evolution due to saturating bond dimensions. This establishes QTT as a powerful tool for nonlinear quantum simulations.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
We thank you and the three reviewers for the careful reading of our manuscript “Solving the Gross–Pitaevskii Equation with Quantic Tensor Trains: Ground States and Nonlinear Dynamics” and for the constructive comments and suggestions. We are pleased that all reviewers found the topic timely and the approach technically solid. We have carefully revised the manuscript in response to all comments, and we believe the new version has been substantially improved in clarity, scope, and scientific positioning.
Below we provide a detailed, point-by-point response to the reviewers’ reports, summarizing all revisions implemented in the manuscript.
List of changes
Major changes: 1. We add a new figure, Fig. 7, for new simulations with a much finer 2^17×2^17 discretization, and up to 125 vortices. Fig.7 shows the vortex density profiles for the systems with different numbers of vortices. 2. We add a new figure, Fig. 8, showing a benchmark of the efficiency of our method compared with several regular finite-difference methods. The system has 7 vortices (see Fig. 7(a)) on a 2^11×2^11 grid. 3. We add a new paragraph in Sec. 4.2 discussing our new benchmark results above. In Sec. 1, Introduction, we add a new paragraph, expand the other corresponding paragraphs, and cite corresponding references to summarize the existing methods for solving the GPE.
Other changes: 4. We update Table 1 for the parameters used in the new simulations. 5. We add Eq. 4 to clearly state the normalization condition we use and refer readers to Appendix A for more details. 6. In Sec. 3, we add a new paragraph indicating the boundary condition and explain why the boundary effect is negligible in our systems. 7. In Sec. 3.1, we add a few sentences and cite an introductory lecture note (Ref. 58) for a more thorough introduction to the TDVP method. 8. In the caption of Fig.2, we clearly state that the error is the absolute error. 9. In Fig.3, we change the definition of the error to the fidelity error 1-|<psipsi_QTT>|^2, and state it in the main text. 10. We add Fig. 13 in Appendix D to discuss convergence with the number of qubits (grid resolution).
Current status:
Reports on this Submission
Strengths
Report
For the ITE, I agree that the explicit methods are easier to implement and the current comparison results are enough to show the advantage of the QTT. However, just as a remark, at least for conventional spatial discretizations, it is nowadays clear that semi-implicit time discretizations perform significantly better than explicit methods as much larger time steps can be used which leads to much faster convergence to the ground states, where the computational cost of solving a linear system is also taken into consideration.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Strengths
Weaknesses
Report
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)
Strengths
Apart from what is mentioned in the last report: 1. Very extensive numerical experiments with novel results.
Weaknesses
- Some additional clarification of the comparison (Fig 8) is needed for readers to have a clear message from this numerical experiment.
Report
(1) I am not sure if I understand what is happening with Fig. 8 for all the "conventional methods" which exhibits O(1) error. Is it that during the CPU runtime that QTT method has converged, these grid-based codes have not done any iterations?
(2) By the way, for a 2^17=131,072 matrix, certainly one should not be storing them as dense matrices. For Gross-Pitaevski matrices the stencils have very good sparse structures.
In any case, I would like some more information in this numerical example.
(3) I am also not sure I understand the choice of "backward" versions of the grid-based methods. Since the QTT algorithm is running gradient descent, shouldn't the comparison just be grid-based methods with gradient descent? Are these grid-based methods running imaginary-time, or even real-time evolution? Even if we are running imaginary-time evolution, we should not be doing the backward version (since QTT is also on the forward level).
But there is perhaps nothing "backward" about what the authors are currently running. It would be good to have some clarifications on this.
In addition, since a particular highlight of this paper is the ability to calculate a large number of vortices, it would be good to add a discussion on the physics implications, including what kind of new physics would happen with a large number of vortices, and connections to experiments.
Requested changes
- Response to the above questions regarding Fig. 8.
- Some physical discussions about the many-vortex regime of BEC.
Recommendation
Ask for minor revision