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 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 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 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.
In the present paper, we propose and analyse a class of tensor methods for the efficient numerical computation of dynamics and spectrum of high-dimensional Hamiltonians. The main focus is on the complex-time evolution problems. We apply the recent quantized-TT (QTT) matrix product states tensor approximation that allows to represent N -d tensors generated by grid representation of d-dimensional functions and operators with log-volume complexity, O(d logN), where N is the univariate discretization parameter in space. Making use of the truncated Cayley transform method allows us to recursively separate the time and space variables and then introduce the efficient QTT representation of both the temporal and spatial parts of solution to the high-dimensional evolution equation. We prove the exponential convergence of the m-term time-space separation scheme and describe the efficient tensor-structured preconditioners for the arising system with multidimensional Hamiltonians. For a class of “analytic” and low QTT-rank input data, our method allows to compute the solution at a fix point in time t = T > 0 with the asymptotic complexity of order O(d logN ln 1e ), where e > 0 is the error bound and q a fixed small number. The time-and-space separation method via the QTT-Cayley-transform enables us to construct the global m-term separable (x, t)-representation of a solution on very fine time-space grid with complexity of order O(dm4 logNt logN), where Nt is the number of sampling points in time. The latter allows the efficient energy spectrum calculations by FFT (or QTTFFT) of autocorrelation function computed on sufficiently long time interval [0, T ]. Moreover, we show that the spectrum of Hamiltonian can also be represented by the poles of the t-Laplace transform of a solution. In particular, the approach can be an option to compute dynamics and spectrum in the time-dependent molecular Schrodinger equation. AMS Subject Classification: 65F30, 65F50, 65N35, 65F10