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The Journal of Chemical Physics

Quantum dynamics of coupled excitons and phonons in chain-like systems: Tensor train approaches and higher-order propagators

Patrick Gelß, Sebastian Matera, Rupert Klein, Burkhard Schmidt

Published Apr 21, 2025• 2 citations
Publisher

Abstract

We investigate tensor-train approaches to the solution of the time-dependent Schrödinger equation for chain-like quantum systems with on-site and nearest-neighbor interactions only. Using efficient low-rank tensor train representations, we aim at reducing memory consumption and computational costs. As an example, coupled excitons and phonons modeled in terms of Fröhlich-Holstein type Hamiltonians are studied here. By comparing our tensor-train-based results with semi-analytical results, we demonstrate the key role of the ranks of the quantum state vectors. Typically, an excellent quality of solutions is found only when the maximum number of ranks exceeds a certain value. One class of propagation schemes builds on splitting the Hamiltonian into two groups of interleaved nearest-neighbor interactions commutating within each of the groups. In particular, the fourth-order Yoshida-Neri and the eighth-order Kahan-Li symplectic composition yield results close to machine precision. Similar results are found for fourth and eighth order global Krylov scheme. However, the computational effort currently restricts the use of these four propagators to rather short chains, which also applies to propagators based on the time-dependent variational principle, typically used for matrix product states. Yet, another class of propagators involves explicit, time-symmetrized Euler integrators. Especially, the fourth-order variant is recommended for quantum simulations of longer chains, even though the high precision of the splitting schemes cannot be reached. Moreover, the scaling of the computational effort with the dimensions of the local Hilbert spaces is much more favorable for the differencing than for splitting or variational schemes.

DOI
10.1063/5.0258904

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We investigate tensor-train approaches to the solution of the time-dependent Schrödinger equation for chain-like quantum systems with on-site and nearest-neighbor interactions only. Using efficient low-rank tensor train representations, we aim at reducing memory consumption and computational costs. As an example, coupled excitons and phonons modeled in terms of Fröhlich-Holstein type Hamiltonians are studied here. By comparing our tensor-train-based results with semi-analytical results, we demonstrate the key role of the ranks of the quantum state vectors. Typically, an excellent quality of solutions is found only when the maximum number of ranks exceeds a certain value. One class of propagation schemes builds on splitting the Hamiltonian into two groups of interleaved nearest-neighbor interactions commutating within each of the groups. In particular, the fourth-order Yoshida-Neri and the eighth-order Kahan-Li symplectic composition yield results close to machine precision. Similar results are found for fourth and eighth order global Krylov scheme. However, the computational effort currently restricts the use of these four propagators to rather short chains, which also applies to propagators based on the time-dependent variational principle, typically used for matrix product states. Yet, another class of propagators involves explicit, time-symmetrized Euler integrators. Especially, the fourth-order variant is recommended for quantum simulations of longer chains, even though the high precision of the splitting schemes cannot be reached. Moreover, the scaling of the computational effort with the dimensions of the local Hilbert spaces is much more favorable for the differencing than for splitting or variational schemes.
We investigate tensor-train approaches to the solution of the time-dependent Schrödinger equation for chain-like quantum systems with on-site and nearest-neighbor interactions only. Using efficient low-rank tensor train representations, we aim at reducing memory consumption and computational costs. As an example, coupled excitons and phonons modeled in terms of Fröhlich-Holstein type Hamiltonians are studied here. By comparing our tensor-train based results with semi-analytical results, we demonstrate the key role of the ranks of the quantum state vectors. Typically, an excellent quality of the solutions is found only when the maximum number of ranks exceeds a certain value. One class of propagation schemes builds on splitting the Hamiltonian into two groups of interleaved nearest-neighbor interactions commutating within each of the groups. In particular, the 4-th order Yoshida-Neri and the 8-th order Kahan-Li symplectic composition yield results close to machine precision. Similar results are found for 4-th and 8-th order global Krylov scheme. However, the computational effort currently restricts the use of these four propagators to rather short chains which also applies to propagators based on the time-dependent variational principle, typically used for matrix product states. Yet another class of propagators involves explicit, time-symmetrized Euler integrators. Especially the 4-th order variant is recommended for quantum simulations of longer chains, even though the high precision of the splitting schemes cannot be reached. Moreover, the scaling of the computational effort with the dimensions of the local Hilbert spaces is much more favorable for the differencing than for splitting or variational schemes.
We demonstrate how to apply the tensor-train format to solve the time-independent Schrödinger equation for quasi-one-dimensional excitonic chain systems with and without periodic boundary conditions. The coupled excitons and phonons are modeled by Fröhlich-Holstein type Hamiltonians with on-site and nearest-neighbor interactions only. We reduce the memory consumption as well as the computational costs significantly by employing efficient decompositions to construct low-rank tensor-train representations, thus mitigating the curse of dimensionality. In order to compute also higher quantum states, we introduce an approach that directly incorporates the Wielandt deflation technique into the alternating linear scheme for the solution of eigenproblems. Besides systems with coupled excitons and phonons, we also investigate uncoupled problems for which (semi-)analytical results exist. There, we find that in the case of homogeneous systems, the tensor-train ranks of state vectors only marginally depend on the chain length, which results in a linear growth of the storage consumption. However, the central processing unit time increases slightly faster with the chain length than the storage consumption because the alternating linear scheme adopted in our work requires more iterations to achieve convergence for longer chains and a given rank. Finally, we demonstrate that the tensor-train approach to the quantum treatment of coupled excitons and phonons makes it possible to directly tackle the phenomenon of mutual self-trapping. We are able to confirm the main results of the Davydov theory, i.e., the dependence of the wave packet width and the corresponding stabilization energy on the exciton-phonon coupling strength, although only for a certain range of that parameter. In future work, our approach will allow calculations also beyond the validity regime of that theory and/or beyond the restrictions of the Fröhlich-Holstein type Hamiltonians.
We study low-rank tensor methods for the numerical solution of Schrödinger'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.
We demonstrate how to apply the tensor-train format to solve the time-independent Schrödinger equation for quasi-one-dimensional excitonic chain systems with and without periodic boundary conditions. The coupled excitons and phonons are modeled by Fröhlich–Holstein type Hamiltonians with on-site and nearest-neighbor interactions only. We reduce the memory consumption as well as the computational costs significantly by employing efficient decompositions to construct low-rank tensor-train representations, thus mitigating the curse of dimensionality. In order to compute also higher quantum states, we introduce an approach that directly incorporates the Wielandt deflation technique into the alternating linear scheme for the solution of eigenproblems. Besides systems with coupled excitons and phonons, we also investigate uncoupled problems for which (semi-)analytical results exist. There, we find that in the case of homogeneous systems, the tensor-train ranks of state vectors only marginally depend on the chain length, which results in a linear growth of the storage consumption. However, the central processing unit time increases slightly faster with the chain length than the storage consumption because the alternating linear scheme adopted in our work requires more iterations to achieve convergence for longer chains and a given rank. Finally, we demonstrate that the tensor-train approach to the quantum treatment of coupled excitons and phonons makes it possible to directly tackle the phenomenon of mutual self-trapping. We are able to confirm the main results of the Davydov theory, i.e., the dependence of the wave packet width and the corresponding stabilization energy on the exciton–phonon coupling strength, although only for a certain range of that parameter. In future work, our approach will allow calculations also beyond the validity regime of that theory and/or beyond the restrictions of the Fröhlich–Holstein type Hamiltonians.
Numerically exact simulations of quantum reaction dynamics, including nonadiabatic effects in excited electronic states, are essential to gain fundamental insights into ultrafast chemical reactivity and rigorous interpretations of molecular spectroscopy. Here, we introduce the tensor-train split-operator KSL (TT-SOKSL) method for quantum simulations in tensor-train (TT)/matrix product state (MPS) representations. TT-SOKSL propagates the quantum state as a tensor train using the Trotter expansion of the time-evolution operator, as in the tensor-train split-operator Fourier transform (TT-SOFT) method. However, the exponential operators of the Trotter expansion are applied using a rank-adaptive TT-KSL scheme instead of using the scaling and squaring approach as in TT-SOFT. We demonstrate the accuracy and efficiency of TT-SOKSL as applied to simulations of the photoisomerization of the retinal chromophore in rhodopsin, including nonadiabatic dynamics at a conical intersection of potential energy surfaces. The quantum evolution is described in full dimensionality by a time-dependent wavepacket evolving according to a two-state 25-dimensional model Hamiltonian. We find that TT-SOKSL converges faster than TT-SOFT with respect to the maximally allowed memory requirement of the tensor-train representation and better preserves the norm of the time-evolving state. When compared to the corresponding simulations based on the TT-KSL method, TT-SOKSL has the advantage of avoiding the need to construct the matrix product state Laplacian by exploiting the linear scaling of multidimensional tensor-train Fourier transforms.
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.