On adaptive time-dependent DMRG based on Runge-Kutta methods

Ulrich Schollwoeck

Weitang Li, Jiajun Ren, Zhigang Shuai

The time dependent density matrix renormalization group (TD-DMRG) has become one of the cutting edge methods of quantum dynamics for complex systems. In this paper, we comparatively study the accuracy of three time evolution schemes in the TD-DMRG, the global propagation and compression method with the Runge-Kutta algorithm (P&C-RK), the time dependent variational principle based methods with the matrix unfolding algorithm (TDVP-MU), and with the projector-splitting algorithm (TDVP-PS), by performing benchmarks on the exciton dynamics of the Fenna-Matthews-Olson complex. We show that TDVP-MU and TDVP-PS yield the same result when the time step size is converged and they are more accurate than P&C-RK4, while TDVP-PS tolerates a larger time step size than TDVP-MU. We further adopt the graphical processing units to accelerate the heavy tensor contractions in the TD-DMRG, and it is able to speed up the TDVP-MU and TDVP-PS schemes by up to 73 times.

Nhat A. Nghiem

We propose a new method for simulating certain type of time-dependent Hamiltonian $H(t) = \sum_{i=1}^m γ_i(t) H_i$ where $γ_i(t)$ (and its higher order derivatives) is bounded, computable function of time $t$, and each $H_i$ is time-independent, and could be efficiently simulated. Our quantum algorithms are based on high-order Runge-Kutta method and forward Euler method, where the time interval is divided into subintervals. Then in an iterative manner, the evolution operator at given time step is built upon the evolution operator at previous time step, utilizing algorithmic operations from the recently introduced quantum singular value transformation framework.

Ulrich Schollwoeck

K. Hallberg

Marc Ferdinand Lensink, Janez Mavri, Herman J. C. Berendsen

A density matrix evolution method [H. J. C. Berendsen and J. Mavri, J. Phys. Chem., 97, 13464 (1993)] to simulate the dynamics of quantum systems embedded in a classical environment is applied to study the inelastic collisions of a classical particle with a five-level quantum harmonic oscillator. We improved the numerical performance by rewriting the Liouville-von Neumann equation in the interaction representation and so eliminated the frequencies of the unperturbed oscillator. Furthermore, replacement of the fixed time step fourth-order Runge-Kutta integrator with an adaptive step size control fourth-order Runge-Kutta resulted in significantly lower computational effort at the same desired accuracy. © 1996 by John Wiley & Sons, Inc.