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arxiv: 2603.13990 · v3 · submitted 2026-03-14 · 🪐 quant-ph · math-ph· math.MP

Dynamical Simulations of Schr\"odinger's Equation via Rank-Adaptive Tensor Decompositions

N. Anders Petersson , Chase Hodges-Heilmann , Stefanie G\"unther This is my paper

Pith reviewed 2026-05-15 11:01 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP
keywords tensor decompositionsSchrödinger equationlow-rank approximationsquantum dynamicstensor trainsTDVPnumerical methodsmany-body systems
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The pith

Rank-adaptive tensor decompositions enable memory-efficient simulations of Schrödinger's equation by exploiting low-rank structure in quantum states.

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

This paper investigates low-rank tensor methods for solving Schrödinger's equation in both time-independent and time-dependent cases. The approaches include the basis update and Galerkin method for tensor trains along with TDVP algorithms, all using truncated singular value decomposition for rank adaptation. These techniques allow compressed representations of partially entangled states, avoiding the exponential memory demands of full state-vector methods. Numerical experiments demonstrate the balance between accuracy and compression, highlighting how time steps and truncation thresholds interact as the number of subsystems increases.

Core claim

All these approaches enable memory efficient representations of partially entangled quantum states and thereby mitigate the exponential cost of conventional state-vector formulations. The rank-adaptivity relies on the truncated singular value decomposition, in which the rank of a matrix is reduced by setting its smallest singular values to zero, based on a threshold parameter that controls the truncation error.

What carries the argument

Rank-adaptive tensor decompositions (tensor trains or Tucker format) with truncated SVD for controlling the approximation rank during time evolution.

Load-bearing premise

The quantum states must remain sufficiently low-rank throughout the simulation for the truncation error to stay small enough to be useful.

What would settle it

If the singular values do not decay rapidly enough in a driven system, the method would require ranks that grow exponentially, erasing the memory advantage.

Figures

Figures reproduced from arXiv: 2603.13990 by Chase Hodges-Heilmann, N. Anders Petersson, Stefanie G\"unther.

Figure 1
Figure 1. Figure 1: Left: A tensor diagram of a tensor train (MPS) of an order-4 tensor of size [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: TDVP: evaluating the action of the effective Hamiltonian on the state in a 4-site system. [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: TDVP-2: evaluating the action of the effective two-site Hamiltonian on the merged (green [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: MPS-BUG: application of Hˆ ef f 2 to Ar r2s (green, top row), giving the slope Hˆ ef f 2 Ar r2s (3 solid legs into empty space, bottom row). where the cores temporarily are assigned doubled bond dimensions, Arkspt1q P C 2bk´1ˆdkˆ2bk , Mrjcspt1q P C 2bjc´1ˆdjˆ2bjc , Brmspt1q P C 2bm´1ˆdmˆ2bm, (5.9) for k P r2, jc ´ 1s and m P rjc ` 1, N ´ 1s. Note that the bond dimensions at both ends of the tensor train ar… view at source ↗
Figure 5
Figure 5. Figure 5: Transverse Ising model with J “ 1 and g “ t0, 0.5u, starting from the ground state, integrated with TDVP-2 and MPS-BUG to time T “ 5. Left: run times as functions of the number of qubits. Right: Maximum bond dimensions. both BUG methods to get similar solution errors. This is because TDVP-2 takes two sub-steps within each full time-step. As the number of time-steps increases, the SVD truncation error accum… view at source ↗
Figure 6
Figure 6. Figure 6: Accuracy and storage requirements as function of the SVD truncation parameter ( [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The magnetization observable in a composite system with [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Real and imaginary parts of the control pulse applied to qubit #1, in the case of [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Solution accuracy (left column) and storage requirements (right column) at the final time [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparing Quandary and TDVP-2 on a state-to-state transformation problem for [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
read the original abstract

We study low-rank tensor methods for the numerical solution of Schr\"odinger's equation with time-independent and explicitly time-dependent Hamiltonians, motivated by large-scale simulations of many-body quantum systems and quantum computing devices subject to time-dependent control pulses. We outline the recent application of the "basis update and Galerkin" (BUG) method for tensor trains, and describe the established TDVP and TDVP-2 algorithms based on the time-dependent variational principle. For comparison, we also consider the BUG method in the Tucker format. All these approaches enable memory efficient representations of partially entangled quantum states and thereby mitigate the exponential cost of conventional state-vector formulations. The rank-adaptivity relies on the truncated singular value decomposition, in which the rank of a matrix is reduced by setting its smallest singular values to zero, based on a threshold parameter that controls the truncation error. Numerical experiments on representative time-independent and time-dependent Hamiltonian models quantify the tradeoff between accuracy and compression across methods, with particular attention to the interplay between the time-step and the truncation threshold, and how the computational effort scales with the number of sub-systems in the quantum system.

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 / 1 minor

Summary. The paper presents low-rank tensor methods (BUG for tensor trains, TDVP, TDVP-2, and Tucker-format BUG) for solving Schrödinger's equation under both time-independent and explicitly time-dependent Hamiltonians. It argues that rank-adaptive truncated SVD enables memory-efficient representations of partially entangled states, thereby mitigating the exponential cost of full state-vector simulations, and supports this with numerical experiments that explore accuracy-compression trade-offs, time-step/truncation interplay, and scaling with subsystem number.

Significance. If the low-rank persistence assumption holds for the targeted regimes, the work would offer practical algorithmic tools for large-scale many-body and quantum-control simulations where conventional methods are intractable, with explicit quantification of compression versus accuracy that could guide method selection in quantum information applications.

major comments (2)
  1. [Abstract] Abstract: the central claim that these methods mitigate exponential cost rests on states remaining sufficiently low-rank under explicit time-dependent driving, yet the manuscript provides no a priori rank-growth estimates, long-time fidelity bounds, or comparisons against exact high-rank references at late times; without these, the observed compression in numerical experiments cannot be confirmed to persist when entanglement grows rapidly.
  2. [Abstract] Abstract: the numerical experiments are described as quantifying the accuracy-compression tradeoff and time-step/truncation interplay, but no error bars, baseline comparisons to full state-vector or other tensor methods, or detailed convergence analysis with respect to truncation threshold are reported, leaving the soundness of the quantified trade-offs unverifiable from the given information.
minor comments (1)
  1. The abstract would benefit from naming the specific Hamiltonian models and system sizes used in the experiments to allow readers to assess the scope of the claimed scaling behavior.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment point by point below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that these methods mitigate exponential cost rests on states remaining sufficiently low-rank under explicit time-dependent driving, yet the manuscript provides no a priori rank-growth estimates, long-time fidelity bounds, or comparisons against exact high-rank references at late times; without these, the observed compression in numerical experiments cannot be confirmed to persist when entanglement grows rapidly.

    Authors: We agree that a priori rank-growth estimates and long-time fidelity bounds would provide stronger support for the central claim. Deriving such general theoretical results for arbitrary time-dependent Hamiltonians remains an open challenge in the literature and is outside the scope of this primarily algorithmic and numerical study. The presented methods adapt ranks dynamically at each step via the SVD threshold to control local truncation error, and the numerical experiments on representative models (both time-independent and time-dependent) demonstrate sustained compression with increasing subsystem number. We will add a dedicated paragraph in the discussion section acknowledging the low-rank persistence assumption, its limitations under rapid entanglement growth, and pointers to related theoretical results on entanglement dynamics. revision: partial

  2. Referee: [Abstract] Abstract: the numerical experiments are described as quantifying the accuracy-compression tradeoff and time-step/truncation interplay, but no error bars, baseline comparisons to full state-vector or other tensor methods, or detailed convergence analysis with respect to truncation threshold are reported, leaving the soundness of the quantified trade-offs unverifiable from the given information.

    Authors: The experiments section already reports accuracy versus compression ratios, time-step/truncation sensitivity, and scaling with subsystem count for the BUG, TDVP, TDVP-2, and Tucker-BUG variants. We acknowledge that the presentation would benefit from explicit error bars (where multiple realizations are feasible), additional baseline comparisons against full state-vector results on smaller systems, and more granular convergence plots versus truncation threshold. We will revise the results section and figures accordingly to include these elements for improved verifiability. revision: yes

Circularity Check

0 steps flagged

No significant circularity in tensor methods for Schrödinger dynamics

full rationale

The paper applies established rank-adaptive tensor techniques (BUG, TDVP, TDVP-2, Tucker) to the Schrödinger equation using standard linear-algebra primitives: truncated SVD for rank reduction and Galerkin projection for time stepping. These operations are independent of the target quantum problem and do not derive their properties from the Schrödinger dynamics itself. Numerical experiments quantify accuracy-compression tradeoffs and scaling with subsystem count, but the low-rank assumption is treated as an empirical precondition rather than a derived result. No self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citations that reduce the central claim to tautology are present. The derivation chain remains self-contained against external benchmarks from tensor algebra.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on the mathematical guarantee that truncated SVD yields the optimal low-rank matrix approximation in the Frobenius norm and on the modeling assumption that quantum states of interest stay compressible under the chosen dynamics.

free parameters (1)
  • truncation threshold
    User-chosen parameter that sets the cutoff for singular values during rank reduction; directly controls the compression-accuracy trade-off.
axioms (1)
  • standard math Truncated singular value decomposition provides the optimal low-rank approximation in the Frobenius norm
    Invoked to justify the rank-adaptive step in BUG and related methods.

pith-pipeline@v0.9.0 · 5508 in / 1315 out tokens · 74876 ms · 2026-05-15T11:01:40.059988+00:00 · methodology

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Works this paper leans on

32 extracted references · 32 canonical work pages

  1. [1]

    Fast and converged classical simulations of evidence for the utility of quantum computing before fault tolerance.Science Advances, 10(3):eadk4321, 2024

    Tomislav Beguˇ si´ c, Johnnie Gray, and Garnet Kin-Lic Chan. Fast and converged classical simulations of evidence for the utility of quantum computing before fault tolerance.Science Advances, 10(3):eadk4321, 2024

    work page 2024

  2. [2]

    Classical solution of the

    Huanchen Zhai, Chenghan Li, Xing Zhang, Zhendong Li, Seunghoon Lee, and Garnet Kin-Lic Chan. Classical solution of the FeMo-cofactor model to chemical accuracy and its implications, 2026. arXiv- 2601.04621

    work page arXiv 2026

  3. [3]

    Daley, C

    A.J. Daley, C. Kollath, U. Schollw¨ ock, and G. Vidal. Time-dependent density-matrix renormalization- group using adaptive effective Hilbert spaces.J. Stat. Mech., 2004. 23

    work page 2004

  4. [4]

    I. V. Oseledets. Tensor-train decomposition.SIAM Journal on Scientific Computing, 33(5):2295–2317, 2011

    work page 2011

  5. [5]

    Ignacio Cirac, Tobias J

    Jutho Haegeman, J. Ignacio Cirac, Tobias J. Osborne, Iztok Piˇ zorn, Henri Verschelde, and Frank Verstraete. Time-dependent variational principle for quantum lattices.Phys. Rev. Lett., 107:070601, Aug 2011

    work page 2011

  6. [6]

    Unifying time evolution and optimization with matrix product states.Phys

    Jutho Haegeman, Christian Lubich, Ivan Oseledets, Bart Vandereycken, and Frank Verstraete. Unifying time evolution and optimization with matrix product states.Phys. Rev. B, 94:165116, Oct 2016

    work page 2016

  7. [7]

    Manmana, Ulrich Schollw¨ ock, and Claudius Hubig

    Sebastian Paeckel, Thomas K¨ ohler, Andreas Swoboda, Salvatore R. Manmana, Ulrich Schollw¨ ock, and Claudius Hubig. Time-evolution methods for matrix-product states.Annals of Physics, 411:167998, 2019

    work page 2019

  8. [8]

    Schr¨ oder, D.H.P

    F.A.Y.N. Schr¨ oder, D.H.P. Turban, A.J. Musser, et al. Tensor network simulation of multi- environmental open quantum dynamics via machine learning and entanglement renormalisation.Nat. Commun., 10:1062, 2019

    work page 2019

  9. [9]

    personal communication

    Dominik Sulz. personal communication

    work page

  10. [10]

    An unconventional robust integrator for dynamical low-rank approximation.BIT Numerical Mathematics, 62(1):23–44, Mar 2022

    Gianluca Ceruti and Christian Lubich. An unconventional robust integrator for dynamical low-rank approximation.BIT Numerical Mathematics, 62(1):23–44, Mar 2022

    work page 2022

  11. [11]

    Time integration of tree tensor networks.SIAM Journal on Numerical Analysis, 59(1):289–313, 2021

    Gianluca Ceruti, Christian Lubich, and Hanna Walach. Time integration of tree tensor networks.SIAM Journal on Numerical Analysis, 59(1):289–313, 2021

    work page 2021

  12. [12]

    Low-rank tree tensor network operators for long- range pairwise interactions.SIAM Journal on Scientific Computing, 47(4):A2248–A2271, 2025

    Gianluca Ceruti, Daniel Kressner, and Dominik Sulz. Low-rank tree tensor network operators for long- range pairwise interactions.SIAM Journal on Scientific Computing, 47(4):A2248–A2271, 2025

    work page 2025

  13. [13]

    Second-order robust parallel integrators for dynamical low-rank approximation.BIT Numerical Mathematics, 65(3):31, Jun 2025

    Jonas Kusch. Second-order robust parallel integrators for dynamical low-rank approximation.BIT Numerical Mathematics, 65(3):31, Jun 2025

    work page 2025

  14. [14]

    Kraus is king: High-order completely positive and trace preserv- ing (CPTP) low rank method for the Lindblad master equation.Journal of Computational Physics, 534:114036, 2025

    Daniel Appel¨ o and Yingda Cheng. Kraus is king: High-order completely positive and trace preserv- ing (CPTP) low rank method for the Lindblad master equation.Journal of Computational Physics, 534:114036, 2025

    work page 2025

  15. [15]

    Spencer Lee and Daniel Appel¨ o. High-order Hermite optimization: Fast and exact gradient computation in open-loop quantum optimal control using a discrete adjoint approach.Journal of Computational Physics, 552:114697, 2026

    work page 2026

  16. [16]

    Anders Petersson, Stefanie G¨ unther, and Seung Whan Chung

    N. Anders Petersson, Stefanie G¨ unther, and Seung Whan Chung. A time-parallel multiple-shooting method for large-scale quantum optimal control.Journal of Computational Physics, 524:113712, 2025

    work page 2025

  17. [17]

    Kolda.Tensor Decompositions for Data Science

    Grey Ballard and Tamara G. Kolda.Tensor Decompositions for Data Science. Cambridge University Press, 2025

    work page 2025

  18. [18]

    Hubig, I

    C. Hubig, I. P. McCulloch, and U. Schollw¨ ock. Generic construction of efficient matrix product opera- tors.Phys. Rev. B, 95:035129, Jan 2017

    work page 2017

  19. [19]

    P. A. M. Dirac. Note on exchange phenomena in the Thomas atom.Mathematical Proceedings of the Cambridge Philosophical Society, 26(3):376–385, 1930

    work page 1930

  20. [20]

    Frenkel.Wave mechanics: advanced general theory.Clarendon Press Oxford, 1964

    J. Frenkel.Wave mechanics: advanced general theory.Clarendon Press Oxford, 1964

    work page 1964

  21. [21]

    McLachlan

    A.D. McLachlan. A variational solution of the time-dependent schrodinger equation.Molecular Physics, 8(1):39–44, 1964

    work page 1964

  22. [22]

    Kramer and M

    P.H. Kramer and M. Saraceno.Geometry of the Time-Dependent Variational Principle in Quantum Mechanics. Springer, 1981

    work page 1981

  23. [23]

    Broeckhove, L

    J. Broeckhove, L. Lathouwers, E. Kesteloot, and P. Van Leuven. On the equivalence of time-dependent variational principles.Chemical Physics Letters, 149(5):547–550, 1988

    work page 1988

  24. [24]

    Oseledets, and Bart Vandereycken

    Christian Lubich, Ivan V. Oseledets, and Bart Vandereycken. Time integration of tensor trains.SIAM Journal on Numerical Analysis, 53(2):917–941, 2015. 24

    work page 2015

  25. [25]

    A parallel basis update and Galerkin integrator for tree tensor networks.SIAM Journal on Scientific Computing, 48(1):A27–A48, 2026

    Gianluca Ceruti, Jonas Kusch, Christian Lubich, and Dominik Sulz. A parallel basis update and Galerkin integrator for tree tensor networks.SIAM Journal on Scientific Computing, 48(1):A27–A48, 2026

    work page 2026

  26. [26]

    A rank-adaptive robust integrator for dynamical low-rank approximation.BIT, 62(4):1149–1174, December 2022

    Gianluca Ceruti, Jonas Kusch, and Christian Lubich. A rank-adaptive robust integrator for dynamical low-rank approximation.BIT, 62(4):1149–1174, December 2022

    work page 2022

  27. [27]

    Monte Carlo wave-function method in quantum optics

    Klaus Mølmer, Yvan Castin, and Jean Dalibard. Monte Carlo wave-function method in quantum optics. J. Opt. Soc. Am. B, 10(3):524–538, Mar 1993

    work page 1993

  28. [28]

    Anders Petersson Stefanie G¨ unther

    N. Anders Petersson Stefanie G¨ unther. Quandary: Optimal control for open and closed quantum systems.https://github.com/LLNL/quandary/, 2025

    work page 2025

  29. [29]

    White, and E

    Matthew Fishman, Steven R. White, and E. Miles Stoudenmire. The iTensor Software Library for Tensor Network Calculations.SciPost Phys. Codebases, page 4, 2022

    work page 2022

  30. [30]

    Efficient higher- order matrix product operators for time evolution.SciPost Phys., 17:135, 2024

    Maarten Van Damme, Jutho Haegeman, Ian McCulloch, and Laurens Vanderstraeten. Efficient higher- order matrix product operators for time evolution.SciPost Phys., 17:135, 2024

    work page 2024

  31. [31]

    Anders Petersson, and Jonathan L

    Stefanie Gunther, N. Anders Petersson, and Jonathan L. DuBois. Quandary: An open-source C++ package for high-performance optimal control of open quantum systems . In2021 IEEE/ACM Second International Workshop on Quantum Computing Software (QCS), pages 88–98, Los Alamitos, CA, USA, November 2021. IEEE Computer Society

    work page 2021

  32. [32]

    Anders Petersson and Fortino Garcia

    N. Anders Petersson and Fortino Garcia. Optimal control of closed quantum systems via B-splines with carrier waves.SIAM Journal on Scientific Computing, 44(6):A3592–A3616, 2022. 25

    work page 2022