Efficient propagation of the hierarchical equations of motion using the matrix product state method

Abstract

We apply the matrix product state (MPS) method to propagate the hierarchical equations of motion (HEOM). It is shown that the MPS approximation works well in different type of problems, including boson and fermion baths. The MPS method based on the time-dependent variational principle is also found to be applicable to HEOM with over one thousand effective modes. Combining the flexibility of the HEOM in defining the effective modes and the efficiency of the MPS method thus may provide a promising tool in simulating quantum dynamics in condensed phases.

Tensor network simulation of multi-environmental open quantum dynamics via machine learning and entanglement renormalisation

“…Our entanglement (von Neumann entropy) analysis of the core tensor and the construction of an optimal three-legged network using ER nodes is based on a clear mathematical framework and might also be applicable to ML-MCTDH problems. Compared with ML-MCTDH and the present method, the recent MPO-based TEMPO algorithm of Strathearn et al appears to require slightly longer computational time and more computational resources for the Ohmic spin-boson model at zero temperature and α ;= 0.5 34 , as do techniques based on hierarchical equations of motion 35 . However, these latter methods are also efficient in treating the effects of finite temperatures and can reach long times without recurrences, which make them very attractive for looking at experimentally relevant reduced system dynamics.…”

“…A powerful approach to this from chemical physics is the Multi-layer Multi-Configurational Time-Dependent Hartree (ML-MCTDH) algorithm 11,18–20 , but, here, we draw upon the deep insights into many-body wave functions and the theory of low-rank tensor approximations that underlie the highly efficient matrix-product state (MPS) and tensor network state (TNS) ansatz used for strongly correlated problems in condensed matter 21–26 . Encouragingly, MPS ansätze has already been shown to provide highly accurate results for toy models of open systems with hundreds of quantised and highly excited vibrations 27–33 , and have also very recently been applied to the equations of motion of reduced density matrices 34,35 . Unfortunately, MPS techniques are mostly effective for 1D systems, placing strong constraints on the types of the environment that can be simulated (vide infra).…”

Density matrix dynamics in twin-formulation: An efficient methodology based on tensor-train representation of reduced equations of motion

“…77 As a concluding remark we would like to point out that HEOM methodology has been very recently implemented within the TT approximation but with a focus on the structure induced by the spectral density variable on the hierarchy. 62 The approach herein developed follows a different route for the numerical implementation of HEOM based on the use of the double space formalism followed by the tensorization of the Liuoville as well as of the relaxation super-operators. One key advantage of our approach is that no restrictions are imposed on the type of system under examination meaning that highly non-linear systems can be analysed as long as the system-bath interaction is linear in the bath degrees of freedom.…”

“…We also point out that this type of representation has very recently been applied to HEOM within the classical density matrix approach. 62 Let us consider a generic expression of a state of a N dimensional quantum system in the form…”

Combined tensor network/cluster expansion method using logic gates: Illustrated for (bi)excitons by a single-layer MoS2 model system

“…The wavefunction of the spin chain is interpreted as a tensor and decomposed in a tensor network such as a MPS (also called a tensor train (TT)). In mathematics and chemistry a new trend uses TT (or other tensor networks) to compress high dimensional tensors regardless, if the tensor represents an actual quantum mechanical wavefunction [40]. Furthermore for solving partial differential equations in real space quantics tensor trains (QTT) were introduced [41][42][43][44][45][46].…”