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.
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 introduce the “tensor-train split-operator Fourier transform” (TT-SOFT) method for simulations of multidimensional nonadiabatic quantum dynamics. TT-SOFT is essentially the grid-based SOFT method implemented in dynamically adaptive tensor-train representations. In the same spirit of all matrix product states, the tensor-train format enables the representation, propagation, and computation of observables of multidimensional wave functions in terms of the grid-based wavepacket tensor components, bypassing the need of actually computing the wave function in its full-rank tensor product grid space. We demonstrate the accuracy and efficiency of the TT-SOFT method as applied to propagation of 24-dimensional wave packets, describing the S 1 / S 2 interconversion dynamics of pyrazine after UV photoexcitation to the S 2 state. Our results show that the TT-SOFT method is a powerful computational approach for simulations of quantum dynamics of polyatomic systems since it avoids the exponential scaling problem of full-rank grid-based representations.
Accurate quantum dynamics simulations of nonadiabatic processes are important for studies of electron transfer, energy transfer, and photochemical reactions in complex systems. In this comparative study, we benchmark various approximate nonadiabatic dynamics methods with mapping variables against numerically exact calculations based on the tensor-train (TT) representation of high-dimensional arrays, including TT-KSL for zero-temperature dynamics and TT-thermofield dynamics for finite-temperature dynamics. The approximate nonadiabatic dynamics methods investigated include mixed quantum-classical Ehrenfest mean-field and fewest-switches surface hopping, linearized semiclassical mapping dynamics, symmetrized quasiclassical dynamics, the spin-mapping method, and extended classical mapping models. Different model systems were evaluated, including the spin-boson model for nonadiabatic dynamics in the condensed phase, the linear vibronic coupling model for electronic transition through conical intersections, the photoisomerization model of retinal, and Tully's one-dimensional scattering models. Our calculations show that the optimal choice of approximate dynamical method is system-specific, and the accuracy is sensitively dependent on the zero-point-energy parameter and the initial sampling strategy for the mapping variables.
We present quantum dynamics of the multi-state harmonic (MSH) model using numerically exact tensor-train (TT)-based calculations. The MSH model provides a general framework for mapping a realistic system onto an effective model Hamiltonian, which is defined in extended spatial dimensions, ensuring consistent reorganization energies between all state pairs. Its analytic structure allows efficient propagation of wavepackets via a rank-adaptive TT-KSL scheme and rigorous finite-temperature dynamics via TT-thermofield dynamics. These exact results are used to benchmark various approximate semiclassical and mixed quantum-classical dynamics, including the linearized semiclassical (LSC), symmetrical quasiclassical, classical mapping models (CMMs), mean-field Ehrenfest, and fewest-switches surface hopping dynamics. We systematically explore the parameter space of the MSH model by changing electronic coupling, reorganization energy, reaction free energy, and the nuclear characteristic frequency. In the adiabatic-inverted regime, strong electronic coupling and low reorganization energy lead all approximate methods to converge with the exact TT results. In contrast, discrepancies emerge in the nonadiabatic or normal regimes, where resolution-of-identity LSC and CMMs provide the reliable predictions. This study establishes the MSH model as a powerful tool for validating nonadiabatic dynamics methods in complex condensed-phase systems.
We develop a dynamical tensor-train method for the Wigner--Fokker--Planck equation arising in continuous-variable open quantum systems. The Wigner function is evolved directly in tensor-train form using a projector-splitting integrator, while the nonlocal Wigner pseudo-differential operator is treated through low-rank representations of the potential difference and matrix-free exponential actions. The resulting method avoids full tensor-product computations in high-dimensional phase space. Numerical experiments in one, two, and three spatial dimensions demonstrate the accuracy and scalability of the method, including simulations in a six-dimensional phase space.
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
Abstract In the present paper, we propose and analyse a class of tensor methods for the efficient numerical computation of the dynamics and spectrum of high-dimensional Hamiltonians. We focus on the complex-time evolution problems. We apply the quantized-TT (QTT) matrix product states type tensor approximation that allows to represent N-d tensors generated by the grid representation of d-dimensional functions and operators with log-volume complexity, O(d log N), 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 the spatial parts of the 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 the class of "analytic" and low QTT-rank input data, our method allows to compute the solution at a fixed point in time t=T>0 with an asymptotic complexity of order O(d log N ln^q (1/ε)), where ε>0 is the error bound and q is a fixed small number. The time-and-space separation method via the QTT-Cayley-transform enables us to construct a global m-term separable (x,t)-representation of the solution on a very fine time-space grid with complexity of order O(dm^4 log N_t log N), where N_t is the number of sampling points in time. The latter allows efficient energy spectrum calculations by FFT (or QTT-FFT) of the autocorrelation function computed on a sufficiently long time interval [0,T]. Moreover, we show that the spectrum of the 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 the dynamics and the spectrum in the time-dependent molecular Schrödinger equation.