We present an ab initio algorithm for quantum dynamics simulations that reformulates the traditional “curse of dimensionality” that plagues all state-of-the-art techniques for solving the time-dependent Schrödinger equation. Using a stochastic wave-function ansatz that is based on a set of interacting single-particle conditional wave functions, we show that the difficulty of the problem becomes dominated by the number of trajectories needed to describe the process, rather than simply the number of degrees of freedom involved. This highly parallelizable technique achieves quantitative accuracy for situations in which mean-field theory drastically fails to capture qualitative aspects of the dynamics, such as quantum decoherence or the reduced nuclear probability density, using orders of magnitude fewer trajectories than a mean-field simulation. We illustrate the performance of this method for two fundamental nonequilibrium processes: a photoexcited proton-coupled electron transfer problem, and nonequilibrium dynamics in a cavity bound electron-photon system in the ultrastrong-coupling regime.
We present an ab initio algorithm for quantum dynamics simulations that reformulates the traditional “curse of dimensionality” that plagues all state-of-the-art techniques for solving the time-dependent Schrödinger equation. Using a stochastic wave-function ansatz that is based on a set of interacting single-particle conditional wave functions, we show that the difficulty of the problem becomes dominated by the number of trajectories needed to describe the process, rather than simply the number of degrees of freedom involved. This highly parallelizable technique achieves quantitative accuracy for situations in which mean-field theory drastically fails to capture qualitative aspects of the dynamics, such as quantum decoherence or the reduced nuclear probability density, using orders of magnitude fewer trajectories than a mean-field simulation. We illustrate the performance of this method for two fundamental nonequilibrium processes: a photoexcited proton-coupled electron transfer problem, and nonequilibrium dynamics in a cavity bound electron-photon system in the ultrastrong-coupling regime.
We present an ab initio algorithm for quantum dynamics simulations that reformulates the traditional ``curse of dimensionality'' that plagues all state-of-the-art techniques for solving the time-dependent Schr\"odinger equation. Using a stochastic wave-function ansatz that is based on a set of interacting single-particle conditional wave functions, we show that the difficulty of the problem becomes dominated by the number of trajectories needed to describe the process, rather than simply the number of degrees of freedom involved. This highly parallelizable technique achieves quantitative accuracy for situations in which mean-field theory drastically fails to capture qualitative aspects of the dynamics, such as quantum decoherence or the reduced nuclear probability density, using orders of magnitude fewer trajectories than a mean-field simulation. We illustrate the performance of this method for two fundamental nonequilibrium processes: a photoexcited proton-coupled electron transfer problem, and nonequilibrium dynamics in a cavity bound electron-photon system in the ultrastrong-coupling regime.
Simulation of quantum dynamics for many-body systems is an open area of research. For interacting many-body quantum systems, the computer memory necessary to perform calculations has an astronomical value, so that approximated models are needed to reduce the required computational resources. A useful approximation that can often be made is that of quantum-classical dynamics, where the majority of the degrees are treated classically, while a few of them must be treated quantum mechanically. When energy is exchanged very quickly between the quantum subsystem and classical environment, the dynamics is nonadiabatic. Most theories for nonadiabatic dynamics are unsatisfactory, as they fail to properly describe the quantum backreaction of the subsystem on the environment. However, an approach based on the quantum-classical Liouville equation solves this problem. Even so, nonadiabatic dynamics is difficult to implement on a computer, and longer simulation times are often inaccessible due to statistical error. There is thus a need for improved algorithms for nonadiabatic dynamics. In this thesis, two algorithms that utilise the quantum-classical Liouville equation will be qualitatively and quantitatively compared. In addition, stochastic sampling schemes for nonadiabatic transitions will be studied, and a new sampling scheme is introduced [D. A. Uken et al., Phys. Rev. E. 88, 033301 (2013)] which proves to have a dramatic advantage over existing techniques, allowing far longer simulation times to be calculated reliably.
ConspectusSimulating molecular dynamics (MD) within a comprehensive quantum framework has been a long-standing challenge in computational chemistry. An exponential scaling of computational cost renders solving the time dependent Schrödinger equation (TDSE) of a molecular Hamiltonian, including both electronic and nuclear degrees of freedom (DOFs), as well as their couplings, infeasible for more than a few DOFs. In the Born-Oppenheimer (BO), or adiabatic, picture, electronic and nuclear parts of the wave function are decoupled and treated separately. Within this framework, the nuclear wave function evolves along potential energy surfaces (PESs) computed as solutions to the electronic Schrödinger equation parametrized in the nuclear DOFs. This approximation, together with increasingly elaborate numerical approaches to solve the nuclear time dependent Schrödinger equation (TDSE), enabled the treatment of up to a few dozens of degrees of freedom (DOFs). However, for particular applications, such as photochemistry, the BO approximation breaks down. In this regime of non-adiabatic dynamics, solving the full molecular problem including electron-nuclear couplings becomes essential, further increasing the complexity of the numerical solution. Although valuable methods such as multiconfigurational time-dependent Hartree (MCTDH) have been proposed for the solution of the coupled electron-nuclear dynamics, they remain hampered by an exponential scaling in the number of nuclear DOFs and by the difficulty of finding universal variational forms.In this Account, we present a perspective on novel quantum computational algorithms, aiming to alleviate the exponential scaling inherent to the simulation of many-body quantum dynamics. In particular, we focus on the derivation and application of quantum algorithms for adiabatic and non-adiabatic quantum dynamics, which include efficient approaches for the calculation of the BO potential energy surfaces (PESs). Thereafter, we study the time-evolution of a model system consisting of two coupled PESs in first and second quantization. In a first application, we discuss a recently introduced quantum algorithm for the evolution of a wavepacket in first quantization and exploit the potential quantum advantage of mapping its spatial grid representation to logarithmically many qubits. For the second demonstration, we move to the second quantization framework and review the scaling properties of two alternative time-evolution algorithms, namely, a variational quantum algorithm (VQA) (based on the McLachlan variational principle) and conventional Trotter-type evolution (based on a Lie-Trotter-Suzuki formula). Both methods clearly demonstrate the potential of quantum algorithms and their favorable scaling compared to the available classical approaches. However, a clear demonstration of quantum advantage in the context of molecular quantum dynamics may require the implementation of these algorithms in fault-tolerant quantum computers, while their application in near-term, noisy quantum devices is still unclear and deserves further investigation.
In this paper we propose an ab initio method to solve quantum many-body problems of molecular dynamics where both electronic and nuclear degrees are represented by ensembles of trajectories and guiding waves in physical space. Both electrons and nuclei can be treated quantum mechanically where the guiding waves obey a set of coupled Schrodinger equations (quantum-quantum description) or, alternatively, coupled Schrodinger-Newtonian equations are solved for the quantum-classical approximation. The method takes into account local and nonlocal quantum correlation effects in a self-consistent manner. The general formalism is applied to one- and two-dimensional hydrogen molecules subjected to a strong ultrashort optical pulse. Comparison is made with the results from the "exact" Ehrenfest molecular dynamics for the molecular ionization and for the evolution of the internuclear distance as the molecule dissociates.
The curse of dimensionality (COD) limits the current state-of-the-art ab initio propagation methods for non-relativistic quantum mechanics to relatively few particles. For stationary structure calculations, the coupled-cluster (CC) method overcomes the COD in the sense that the method scales polynomially with the number of particles while still being size-consistent and extensive. We generalize the CC method to the time domain while allowing the single-particle functions to vary in an adaptive fashion as well, thereby creating a highly flexible, polynomially scaling approximation to the time-dependent Schrödinger equation. The method inherits size-consistency and extensivity from the CC method. The method is dubbed orbital-adaptive time-dependent coupled-cluster, and is a hierarchy of approximations to the now standard multi-configurational time-dependent Hartree method for fermions. A numerical experiment is also given.
We present a semiclassical approach for nonadiabatic molecular dynamics based on the Ehrenfest method with corrections for decoherence and detailed balance. Decoherence is described via a coherence penalty functional that drives dynamics away from regions in Hilbert space characterized by large values of coherences. Detailed balance is incorporated by modification of the off-diagonal matrix elements with a quantum correction factor used in semiclassical approximations to quantum time-correlation functions. Both decoherence and detailed balance corrections introduce nonlinear terms to the Schrödinger equation. At the same time, the simplicity of fully deterministic dynamics and a single trajectory for each initial condition is preserved. In contrast, surface hopping is stochastic and requires averaging over multiple realization of the stochastic process for each initial condition. The Ehrenfest-decoherence-detailed-balance (Ehrenfest-DDB) method is adapted to the classical path approximation and ab initio time-dependent density functional theory and applied to an experimentally studied nanoscale system consisting of a fluorophore molecule and an scanning tunneling microscopy tip and undergoing current-induced charge injection, cooling, and recombination. Ehrenfest-DDB produces time scales that are similar to those obtained with decoherence induced surface hopping, which is a popular nonadiabatic molecular dynamics technique applied to condensed matter. At long times, Ehrenfest-DDB dynamics slows down considerably because the detailed balance correction makes off-diagonal elements go to zero on approach to Boltzmann equilibrium. The Ehrenfest-DDB technique provides efficient means to study quantum dynamics in large systems.