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 and efficiency for three time evolution schemes in TD-DMRG method, the direct propagation and compression with 4th order Runge-Kutta propagator (P&C-RK4), time dependent variational principle with matrix unfolding (TDVP-MU) and projector splitting (PS) by performing benchmarks on the exciton dynamics of Fenna-Matthews-Olson (FMO) complex. We show that the PS scheme is the most accurate and the fastest method because the propagation in each step is unitary by combining Krylov subspace algorithm and there is no error caused by regularization or instability of matrix inversion. We also compare the parallelization efficiency of multi-core central processing units (CPU) with graphical processing units (GPU) and we find that although 28 cores of CPU can merely double the speed, GPU is able to speed up the TDVP-MU and PS scheme by up to 57 times through efficient acceleration of matrix multiplication. Using the optimal strategy, we are able to simulate the full quantum exciton dynamics of 7-site FMO model with 252 degrees of freedom to a timescale of 1 ps at fairly high accuracy within only 4 minutes.
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 and efficiency for three time evolution schemes in TD-DMRG method, the direct propagation and compression with 4th order Runge-Kutta propagator (P&C-RK4), time dependent variational principle with matrix unfolding (TDVP-MU) and projector splitting (PS) by performing benchmarks on the exciton dynamics of Fenna-Matthews-Olson (FMO) complex. We show that the PS scheme is the most accurate and the fastest method because the propagation in each step is unitary by combining Krylov subspace algorithm and there is no error caused by regularization or instability of matrix inversion. We also compare the parallelization efficiency of multi-core central processing units (CPU) with graphical processing units (GPU) and we find that although 28 cores of CPU can merely double the speed, GPU is able to speed up the TDVP-MU and PS scheme by up to 57 times through efficient acceleration of matrix multiplication. Using the optimal strategy, we are able to simulate the full quantum exciton dynamics of 7-site FMO model with 252 degrees of freedom to a timescale of 1 ps at fairly high accuracy within only 4 minutes.
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.
We propose a new heterogeneous parallel strategy for the density matrix renormalization group (DMRG) method in the hybrid architecture with both central processing unit (CPU) and graphics processing unit (GPU). Focusing on the two most time-consuming sections in the finite DMRG sweeps, i.e., the diagonalization of superblock and the truncation of subblock, we optimize our previous hybrid algorithm to achieve better performance. For the former, we adopt OpenMP application programming interface on CPU and use our own subroutines with higher bandwidth on GPU. For the later, we use GPU to accelerate matrix and vector operations involving the reduced density matrix. Applying the parallel scheme to the Hubbard model with next-nearest hopping on the 4-leg ladder, we compute the ground state of the system and obtain the charge stripe pattern which is usually observed in high temperature superconductors. Based on simulations with different numbers of DMRG kept states, we show significant performance improvement and computational time reduction with the optimized parallel algorithm. Our hybrid parallel strategy with superiority in solving the ground state of quasi-two dimensional lattices is also expected to be useful for other DMRG applications with large numbers of kept states, e.g., the time dependent DMRG algorithms.
Density matrix renormalization group (DMRG), as a numerical method of solving the ground state of one-dimensional strongly-correlated lattice model with very high accuracy, requires expensive computational and memory cost when applied to two- and quasi-two-dimensional problems. The number of DMRG kept states is generally very large to achieve a reliable accuracy for these applications, which results in numerous matrix and vector operations and unbearably consuming time in the absence of the proper parallelization. However, due to its sequential nature, the parallelization of DMRG algorithm is usually not straightforward. In this work, we propose a new hybrid parallelization strategy for the DMRG method. It takes advantage of the computing capability of both central processing unit (CPU) and graphics processing unit (GPU) of the computer. In order to achieve as many as DMRG kept states within a limited GPU memory, we adopt the four-block formulation of the Hamiltonian rather than the two-block formulation. The later consumes much more memories, which has been used in another pioneer work on the hybrid parallelization of the DMRG algorithm, and only a small number of DMRG kept states are available. Our parallel strategy focuses on the diagonalization of the Hamiltonian, which is the most time-consuming part of the whole DMRG procedure. A hybrid parallelization strategy of diagonalization method is implemented, in which the required data for diagonalization are distributed on both the host and GPU memory, and the data exchange between them is negligible in our data partitioning scheme. The matrix operations are also shared on both CPU and GPU when the Hamiltonian acts on a wave function, while the distribution of these operations is determined by a load balancing strategy. Taking fermionic Hubbard model for example, we examine the running performance of the hybrid parallelization strategy with different DMRG kept states and provide corresponding performance benchmark. On a 4-leg ladder, we employ the conserved quantities with U(1) symmetry of the model and a good-quantum number based task scheduling to further reduce the GPU memory cost. We manage to obtain a moderate speedup of the hybrid parallelization for a wide range of DMRG kept states. In our example, the ground state energy with high accuracy is obtained by the extrapolation of the results, with different numbers of states kept, and we show charge stripes which are usually experimentally observed in high-temperature superconductors. In this case, we keep 104 DMRG states and the GPU memory cost is less than 12 Gigabytes.
Recent advances in experimental techniques have enabled precise characterization of fundamental nonadiabatic and even electronic dynamics in molecules, on ultrafast time scales reaching femtosecond to attosecond resolution. However, accurate theoretical simulation of these ultrafast chemical dynamics processes in large systems remains challenging, largely due to the overwhelming number of degrees of freedom (DoFs) and the pronounced many-body correlations. In recent years, by leveraging efficient decomposition schemes for high-dimensional wave function and operator tensors, the time-dependent density matrix renormalization group (TD-DMRG) has emerged as a powerful and accurate quantum dynamics method for simulating nonadiabatic and electron dynamics in large chemical systems, in conjunction with realistic electron/exciton-vibration/phonon models or ab initio quantum chemistry many-electron Hamiltonians. This review outlines the fundamentals of TD-DMRG for chemical dynamics, covering matrix product state/operator (MPS/MPO) frameworks and algorithms from ground-state calculations to time evolution. We discuss thermal/environmental effects and compare TD-DMRG with other tensor network methods such as multiconfiguration time-dependent Hartree (MCTDH) and multilayer MCTDH (ML-MCTDH). Demonstrated applications include simulations of pyrazine absorption, singlet fission in rubrene crystal, and charge migration in chloroacetylene cation. These show TD-DMRG’s capability for modeling complex ultrafast processes from femtoseconds to attoseconds with controlled accuracy.
Combining the time-dependent variational principle (TDVP) algorithm with the parallelization scheme introduced by Stoudenmire and White for the density matrix renormalization group (DMRG), we present the first parallel matrix product state (MPS) algorithm capable of time evolving one-dimensional (1D) quantum lattice systems with long-range interactions. We benchmark the accuracy and performance of the algorithm by simulating quenches in the long-range Ising and XY models. We show that our code scales well up to 32 processes, with parallel efficiencies as high as 86. Finally, we calculate the dynamical correlation function of a 201-site Heisenberg XXX spin chain with 1/r2 interactions, which is challenging to compute sequentially. These results pave the way for the application of tensor networks to increasingly complex many-body systems.
Electronic and/or vibronic coherence has been found by recent ultrafast spectroscopy experiments in many chemical, biological, and material systems. This indicates that there are strong and complicated interactions between electronic states and vibration modes in realistic chemical systems. Therefore, simulations of quantum dynamics with a large number of electronic and vibrational degrees of freedom are highly desirable. Due to the efficient compression and localized representation of quantum states in the matrix-product state (MPS) formulation, time-evolution methods based on the MPS framework, which we summarily refer to as tDMRG (time-dependent density-matrix renormalization group) methods, are considered to be promising candidates to study the quantum dynamics of realistic chemical systems. In this work, we benchmark the performances of four different tDMRG methods, including global Taylor, global Krylov, and local one-site and two-site time-dependent variational principles (1TDVP and 2TDVP), with a comparison to multiconfiguration time-dependent Hartree and experimental results. Two typical chemical systems of internal conversion and singlet fission are investigated: one containing strong and high-order local and nonlocal electron-vibration couplings and the other exhibiting a continuous phonon bath. The comparison shows that the tDMRG methods (particularly, the 2TDVP method) can describe the full quantum dynamics in large chemical systems accurately and efficiently. Several key parameters in the tDMRG calculation including the truncation error threshold, time interval, and ordering of local sites were also investigated to strike the balance between efficiency and accuracy of results.