Solving the Gross-Pitaevskii Equation with Quantic Tensor Trains: Ground States and Nonlinear Dynamics

• The method tackles a nonlinear quantum PDE with broad applications in physics.
• The QTT framework allows exponential compression, enabling simulations with high resolution at reduced computational cost.
• The variational method shows superior performance over TDVP in convergence, which is essential for nonlinear solvers.
• Benchmark results include challenging systems (e.g., 61-vortex states), which suggests scalability.

• The comparison is only made with the most naive baseline solver. While this is understandable at this early stage of development, the authors should clarify this limitation explicitly in the text. As Referee 2 points out, this method is not yet compared to the current state-of-the-art. However, it is reasonable not to expect a newly proposed QTT approach to outperform highly optimized traditional solvers with a long development history.

Below are detailed suggestions:

• In Fig. 2, the vertical axis is labeled as an error, but it is unclear whether this represents absolute or relative error. Please clarify.
• In Fig. 3, it would help readability to add a note like "See also Fig. 7 for the 7, 19, 37 vortex density plots".
• In Fig. 4, while the QTT-based scaling is visually compared to grid-based methods, a brief note on the number of discretization points (qubits) needed for sufficient accuracy would make the comparison more balanced.
• In Fig. 4’s legend, the bond dimension D should be italicized for consistency.
• Appendix A is not cited in the main text. It discusses the role of normalization in the variational method, but it is unclear whether strict normalization is essential or whether the observed difficulties also appear without normalization.

Technical definitions (should be clarified in the revised manuscript): - The QTT compression error is defined as 1 - |<ψ|ψ_QTT>|, but a more geometric error would be |1 - <ψ|ψ_QTT>|, which satisfies ‖ψ - ψ_QTT‖ ~ O(δ^2). It would help to comment on this, as some tensor network literature (e.g., ITensor) uses the latter form for truncation criteria. - In Fig. 8, the error measure ε_ψ² is not defined in the caption or main text. It should be stated explicitly whether it refers to the Frobenius norm ‖ψ - ψ_QTT‖_F or its square.

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This paper provides a complete exploration of using the QTT format for solving GPE. The following things are clearly established: 1. The nonlinearity in QPE could be handled by QTT. 2. The bond dimension saturates and are moderate.

However, this paper comes from a QTT / MPS perspective. My biggest concern is that it does not seem to sufficiently acknowledge or situate itself within the long list of efforts to solve 2D and 3D Gross-Pitaevski equations, both for stationary states and dynamics, which starts from early 2000s and still continues to today. (For example, a quick search leads me to a review paper in DOI.2025 10.1137/22M1516324.)

1. The current introduction section focus almost entirely on MPS / TT. I strongly recommend that the authors add at least one paragraph summarizing the well-established algorithmic developments for the GPE over the past two decades, such as for spatial discretizations (finite different, finite element, Fourier spectral methods, adaptive meshes etc), minimization algorithms (Riemannian optimzation, and the recent Sobolev gradient descent, just to name a few), and also the eigenvector-based perspective for stationary states.

2. The manuscript repeatedly refers to “conventional” or “traditional” methods. I'd appreciate some clarification on this. For example, are these finite differences/finite elements implementation? How are the boundary conditions handled? Since these methods are used as benchmarks, a more detailed description is essential.

3. I'd like to point out that a comparison to a regular finite-difference/finite-element method for arguing the efficiency of the QTT approach is not necessarily a fair comparison. There exist well-developed improvements within grid-based frameworks — e.g., multigrid solvers and adaptive meshes — which are widely used for a few decades. One could even argue that finite-element methods could deal naturally accommodate irregular geometries while it is not clear how QTT could do that.

This is not to say that the authors are obligated to do comparisons with adaptive grids / multgrid methods, but they still need to fairly address this perspective. Moreover, if they do such comparisons, this paper will be much more convincing scientifically.

1. As a result of Point 3, I invite the authors to reexamine and reconsider some of the arguments they make, about the advantages of the QTT approach. For example, here is a paper from 2006 (DOI. 10.1137/050629392), which calculates rotating BEC with comparable number of vortices to this paper. This may motivate the authors to reevaluate how the contributions of this paper should be described. Positioning the QTT approach as one among several methods, rather than as a categorical improvement over “conventional” techniques, might be more suitable.

I want to conclude by saying that my comments are not meant to discourage the pursuit of new algorithms for classical problems. On the contrary, I believe such efforts are very valuable, provided they are put into proper context and compared carefully with existing approaches.

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The authors propose a tensor method based on the quantic tensor train for the spatial discretization of the GPE. The computation of both ground states and dynamics is considered. Numerical experiments are presented to demonstrate the high accuracy of the proposed method, and the reduced computational cost compared to the grid based methods.

Overall, the manuscript presents a novel and interesting method. However, I have several concerns about the paper.

Main concern: The numerical examples are too simple and thus are not very convincing. From the reviewer's perspective, all the presented numerical results (including the cases of many vortices) can be obtained by using the Fourier pseudospectral method in space [2,3,4]. I would expect more examples that clearly cannot be obtained by the classical grid-based methods.

Other comments:

1. P5. ``The time step is set to be $0.5 \delta x^2$ to satisfy the CFL condition". In fact, by using the semi-implicit discretization in the ITE (see [1]), one no longer need such CFL condition and the convergence of the ITE could be significantly accelerated.

On the other hand, many more efficient methods than the ITE are proposed for computing the ground states, e.g. [2,3,4]. Some discussion on whether the existing methods proposed for the grid-based methods can be transferred to the tensor framework would be very helpful.

1. For the computation of dynamics. The proposed method is a spatial discretization. There are many standard temporal discretizations available in the literature, such as the splitting methods, exponential integrators, Crank-Nicolson methods, and Runge-Kutta methods. Can the proposed method be efficiently coupled with some of these temporal discretizations?

In addition, there are some challenges in the computation of dynamics for standard numerical methods, such as the simulation in the whole space instead of a truncated bounded domain, the simulation of low regularity solutions or solutions with high oscillation in space. Will the proposed methods be (potentially) advantageous in these problems?

1. The numerical schemes in section 3, as well as the schemes used for comparison, are not clearly given in the paper, which should be revised.

References [1] W. Bao, Q. Du, Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25 (2004), pp. 1674--1697.

[2] I. Danaila, B. Protas, Computation of Ground States of the Gross-Pitaevskii Functional via Riemannian Optimization, SIAM J. Sci. Comput., 39 (2017), pp. B1102--B1129.

[3] J. Gaidamour, Q. Tang, X. Antoine, BEC2HPC: A HPC spectral solver for nonlinear Schr\"odinger and rotating Gross-Pitaevskii equations. Stationary states computation, Comput. Phys. Commun., 265 (2021), p. 108007.

[4] H. Chen, G. Dong, W. Liu, Z. Xie, Second-order flows for computing the ground states of rotating Bose-Einstein condensates, J. Comput. Phys., 475 (2023), p. 111872.

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