Functional Tensor-Train Chebyshev Method for Multidimensional Quantum Dynamics Simulations

Abstract

Methods for efficient simulations of multidimensional quantum dynamics are essential for theoretical studies of chemical systems where quantum effects are important, such as those involving rearrangements of protons or electronic configurations. Here, we introduce the functional tensor-train Chebyshev (FTTC) method for rigorous nuclear quantum dynamics simulations. FTTC is essentially the Chebyshev propagation scheme applied to the initial state represented in a continuous analogue tensor-train format. We demonstrate the capabilities of FTTC as applied to simulations of proton quantum dynamics in a 50-dimensional model of hydrogen-bonded DNA base pairs.

Tensor-Train Split Operator KSL (TT-SOKSL) Method for Quantum Dynamics Simulations

“…Finally, we note that the strategy of TT-SOKSL -combining the split-operator Hamiltonian and the efficient KSL projection scheme-could be exploited in other quantum propagation methods. For example, in Chebyshev propagation, 71,76,77 the Hamiltonian is represented via the Chebyshev expansion, and the Chebyshev polynomials can be represented as a tensor trains. 71 So, the TT-SOKSL splitting could provide a more effective scheme to reduce the TT-rank of the Hamiltonian.…”

“…For example, in Chebyshev propagation, 71,76,77 the Hamiltonian is represented via the Chebyshev expansion, and the Chebyshev polynomials can be represented as a tensor trains. 71 So, the TT-SOKSL splitting could provide a more effective scheme to reduce the TT-rank of the Hamiltonian. Solving the split equation with the TT-KSL scheme could provide further computational advantage such as norm conservation and efficient utilization of a low-rank tensor-train array.…”

“…TT-SOKSL relies on the tensor-train (TT) format, 61-64 also called matrix product states (MPS) with open boundary conditions, [65][66][67][68][69] recently explored for the development of methods for quantum dynamics and global optimization. 35,70,71 The TT format of an arbitrary d-dimensional tensor X ∈ C n 1 ×...×n d involves a train-like matrix product of d 3-mode tensors X i ∈ C r i−1 ×n i ×r i with r 0 = r d = 1, so any element X(j 1 , ..., j d ) of X can be evaluated, as follows: 61…”

Functional Tensor-Train Chebyshev Method for Multidimensional Quantum Dynamics Simulations

“…The codes are available in public domain. 86 The minimal and maximal potential energy surface values required for rescaling of the Hamiltonian in the Chebyshev scheme are determined either analytically or through constrained nonlinear optimization to avoid calculation of the multidimensional potential energy surface at all position space grid points considered. Individual Chebyshev polynomials are determined as either tensor-trains or function-trains via the recurrence relation Eq.…”

Ultrafast Charge Relocation Dynamics in Enol–Keto Tautomerization Monitored with a Local Soft-X-ray Probe

“…A rigorous interpretation of the nuclear and electronic rearrangements responsible for changes in the XAS spectra can be provided by simulations of quantum dynamics as described by the tensor-train split-operator Fourier transform (TT-SOFT) method . TT-SOFT is a rigorous method for simulations of quantum dynamics that exploits matrix product-state representations analogous to those employed by recently developed methods for simulating vibrational and fluorescence spectroscopy − and other methods for simulations of quantum dynamics and global optimization. , In TT-SOFT, the initial nuclear wave function evolves according to the Schrödinger equation, and therefore, nuclear quantum effects, such as zero-point energy, delocalization, and interference effects, are naturally incorporated into the simulations. Whereas MCTDH tensor network quantum dynamics relies on the hierarchical Tucker format and MCTDH equations of motion, TT-SOFT employs tensor trains and split-operator Fourier transform dynamics that facilitate simulation of highly multidimensional chemical systems and avoid ill-conditioned matrices (see also ref ).…”