We consider computational methods for simulating the dynamics of molecular systems governed by the time-dependent Schrödinger equation. Solving the Schrödinger equation numerically poses a challenge due to its often highly oscillatory solutions, and to the exponential growth of work and memory with the number of particles in the system. Two different classes of problems are studied: the dynamics of the nuclei in a molecule and the dynamics of an electron in orbit around a nucleus. For the first class of problems we present new computational methods which exploit the relation between quantum and classical dynamics in order to make the computations more efficient. For the second class of problems, the lack of regularity in the solution poses a computational challenge. Using knowledge of the non-smooth features of the solution we construct a new method with two orders higher accuracy than what is achieved by direct application of a difference stencil.
We consider computational methods for simulating the dynamics of molecular systems governed by the time-dependent Schrödinger equation. Solving the Schrödinger equation numerically poses a challenge due to its often highly oscillatory solutions, and to the exponential growth of work and memory with the number of particles in the system. Two different classes of problems are studied: the dynamics of the nuclei in a molecule and the dynamics of an electron in orbit around a nucleus. For the first class of problems we present new computational methods which exploit the relation between quantum and classical dynamics in order to make the computations more efficient. For the second class of problems, the lack of regularity in the solution poses a computational challenge. Using knowledge of the non-smooth features of the solution we construct a new method with two orders higher accuracy than what is achieved by direct application of a difference stencil.
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.
Chemical dynamics, in principle, should be understood by solving the time-dependent Schrödinger equation for a molecular system, describing motion of the nuclei and electrons. However, the computational efforts to solve this partial second-order differential equation scales exponentially with the system size, which prevents us from getting exact numerical solutions for systems larger than 4-5 atoms. Thus, approximations simplifying the picture are necessary. The so-called Born-Oppenheimer approximation, separating motion of the electrons and nuclei is the central one: solution to the electronic Schrödinger equation defines the potential energy surface on which the nuclear motion unfolds, and there are standard quantum chemistry software packages for solving the electronic Schrödinger equation. For the nuclear Schrödinger equation, however, there are no widely applicable quantum-mechanical approaches, and most simulations are performed using classical Newtonian mechanics which is often adequate due to large nuclear masses. However, the nuclear quantum effects are significant for chemical processes involving light nuclei at low energies, and including these effects into simulation, even approximately, is highly desirable. In this dissertation, an approximate methodology of including quantum-mechanical effects within the quantum trajectory or the de Broglie-Bohm formulation of the Schrödinger equations is developed. Use of the trajectory framework makes the approach scalable to hundreds of degrees of freedom. The methodology is applied to study high-dimensional systems (solid He4 and others) relevant to chemistry.
Solving the time-dependent Schrödinger equation is an important application area for quantum algorithms. We consider Schrödinger's equation in the semi-classical regime. Here the solutions exhibit strong multiple-scale behavior due to a small parameter ℏ , in the sense that the dynamics of the quantum states and the induced observables can occur on different spatial and temporal scales. Such a Schrödinger equation finds many applications, including in Born-Oppenheimer molecular dynamics and Ehrenfest dynamics. This paper considers quantum analogues of pseudo-spectral (PS) methods on classical computers. Estimates on the gate counts in terms of ℏ and the precision ε are obtained. It is found that the number of required qubits, m , scales only logarithmically with respect to ℏ . When the solution has bounded derivatives up to order ℓ , the symmetric Trotting method has gate complexity O ( ( ε ℏ ) − 1 2 p o l y l o g ( ε − 3 2 ℓ ℏ − 1 − 1 2 ℓ ) ) , provided that the diagonal unitary operators in the pseudo-spectral methods can be implemented with p o l y ( m ) operations. When physical observables are the desired outcomes, however, the step size in the time integration can be chosen independently of ℏ . The gate complexity in this case is reduced to O ( ε − 1 2 p o l y l o g ( ε − 3 2 ℓ ℏ − 1 ) ) , with ℓ again indicating the smoothness of the solution.
This talk is about a method for solving the numerical initial value problem of the time-dependent Schrodinger equation with a Hamiltonian which may depend on time. It is only for time independent Hamiltonians that I was able to describe the accuracy and stability properties in the form of proven theorems. All practical experience shows, that the time-dependent case behaves equally well. In this method, the Hamiltonian comes in only as a hermitian linear operator and not as a mathematical expression that can be reordered or otherwise transformed in order to speed up the necessary calculations. In all cases where it is a feasible computational task to apply the Hamilton operator to a generic state vector, my suggestion is to try this method as a workhorse. It can be considered a ‘Simpson’s rule’ for quantum dynamics. There is no way for me to explain the matter from a technical point of view in 10 minutes, so I’ll try to convey the basic idea of the method. This idea is to treat quantum dynamics in closer analogy to classical dynamics as one usually does. One may argue, that there is a fundamental difference between classical and quantum dynamics since the Schrodinger equation is first order with respect to the time derivative, whereas classical mechanics is inherently second order, since forces determine accelerations and not velocities. This is not so clear a distinction as one might think, since also classical mechanics can be given the first-order form of canonical equations. I propose the opposite of such an approach, namely to give quantum dynamics a second-order form by simply differentiating the time-dependent Schrodinger equation with respect to time. To be sure, this is only a heuristic device for defining a computationally feasible approximation and not a proposal to change quantum mechanics. For the resulting equation for the second time derivative of the wave function, I then carry over and use the integration methods from the field of granular matter dynamics which I have been using in an industrial project concerning simulation of the toning process in electro-photographic copiers/printers over some years [4]. Such methods come in
This bachelor thesis deals with the time-independent Schrodinger equation and the two ways of solving
it: analytical and numerical. The goal is to show whether the equation can be solved precisely enough
numerically on the computer, which would allow solutions for complicated cases that don’t have
analytical solutions. Specifically, a solution for a particle in a harmonic potential is discussed.
Dependence of the accuracy of the numerical solution on the parameters in the method itself is also
discussed. This work presents an algorithm for the numerical solution of the Schrodinger equation. With
the growth of computer power in the future, we will definitely be able to improve both the speed and
precision of solving even more complicated problems than the one depicted here.