TY  - GEN
AV  - public
CY  - Heidelberg
TI  - Quantum Dynamics of Chemical Systems with Large Number of Degrees of Freedom: Linearized Phase Space Methods and Quantum Simulations
Y1  - 2022///
ID  - heidok32348
A1  - Lang, Haifeng
UR  - https://archiv.ub.uni-heidelberg.de/volltextserver/32348/
N2  - Nonadiabatic quantum dynamics plays an important role in a wide range of chemical reactions and femtochemistry experiments. However, numerically converged simulations are typically only affordable for small size systems because the computational efforts generically increase exponentially. This thesis is devoted to the theoretical analysis of two candidates of simulation methods for large size systems, linearized phase space methods and quantum simulations.


Linearized phase space methods, for instance, fully linearized methods, partially linearized methods, and symmetrical quasi-classcial windowing,  approximate the quantum dynamics as the classical dynamics, and quantum effects are accounted for by Monte Carlo sampling of the initial quantum phase space. The major drawback is that the sampling of independent phase space trajectories neglects quantum coherence and interference. For condensed phase simulations, this limitation fortunately is only minor. Different linearized phase space methods are mainly characterized by the initial electronic phase space selections, and it is believed that the choice of electronic phase space determines the accuracy of the method. While there are lots of numerical results to support this argument, a rigorous theoretical analysis is still outstanding. Rewriting fully and partially linearized methods in a unified expression, we establish a rigorous measure of the short-time accuracy, the intra-electron correlation, which has a close connection to the initial electronic phase space. The methods with correct intra-electron correlation are more accurate in the short-time region for various chemical motivated models than the methods with a wrong one. For various popular linearized phase space methods, including many fully and partially linearzied methods, we also give either a proof of correct intra-electron correlation sampling or an explicit violation example. Our theoretical analysis gives an explanation of the accuracy order of linearized phase space methods reported in the literature. Moreover, the intra-electron correlation can be a guideline for the development future linearized phase space methods.

Further, we introduce the generalized discrete truncated Wigner approximation (GDTWA), which is a well-established linearized phase space method in the field of quantum lattice models, into chemistry. The GDTWA uses the Wootters' discrete phase space for electrons, which can sample the intra-electron correlation correctly for diagonals states. We reformulate the GDTWA in the unified expression of linearized phase space methods, which shows that the GDTWA is a fully--partially hybrid method. With the help of this reformulation, we 
not only reduce the computational efforts, but also demonstrate a reduced zero-point energy accounting without an explicit zero-point energy parameter in the GDTWA. Numerical benchmarks on scattering models and linear vibronic coupling models show a robust performance on various chemical motivated models. Also, we develop two GDTWA, approach I and II, for a particle in gauge vector potentials. Theoretical analysis shows that the two approaches favor the simulation of synthetic gauge field and on-the-fly simulation of molecular dynamics in the adiabatic representation, respectively. Our numerical results of ultracold atoms, linear vibrionic coupling models as synthetic gauge fields, as well as on-the-fly simulations of linear vibronic coupling models confirm the analysis. 

To overcome the difficulty of simulating quantum mechanics arising from exponentially increasing Hilbert space, quantum simulations use controllable quantum devices which obey the rule of quantum mechanics. Nowadays, the imperfect controls of quantum devices have a huge impact on the accuracy of the simulations. Specifically, when the errors of the implementations break symmetries of the system, simulation results could even be qualitatively wrong. We rigorously develop an experimentally feasible linear penalty method to suppress the symmetry-breaking errors. Numerical benchmarks of the lattice gauge theory and the hydrogen molecule show good performances on protections of symmetries, local observables, and wave functions. 

Our theoretical analysis on both linearized phase space methods and quantum simulations illustrate the possibilities of simulating large size systems with large potential for applications in quantum chemistry and related areas.
ER  -