Simultaneous integration of mixed quantum-classical systems by density matrix evolution equations using interaction representation and adaptive time step integrator
Marc Ferdinand Lensink, Janez Mavri, Herman J. C. Berendsen
A density matrix evolution (DME) method (Berendsen, H. J. C., and Mavri, J., 1993, J. phys. Chem., 97, 13464) to simulate the dynamics of quantum systems embedded in a classical environment is applied to study the inelastic collisions of a classical particle with a five level quantum harmonic oscillator. The results are compared with the exact quantum calculations of Secrest, D., and Johnson, B. R., 1966, J. chem. Phys., 45, 4556, and with the results of classical mechanics. The DME results are between the results of the full quantum treatment and classical mechanics but closer to the latter. Furthermore, we demonstrate the time reversibility of the DME equations of motion and discuss the importance for the time reversibility of the quantum phase described by the off-diagonal elements of the density matrix.
A density matrix evolution(DME) method to simulate the dynamics of quantum systems embedded in a classical environment is presented. The method is applicable when the quantum dynamical degrees of freedom can be described in a Hilbert space of limited dimensionality. The method is applied to the case of proton-transfer reactions in a fluctuating double-well potential [details are given in the following paper in this issue] and compared to other analytical and numerical solutions. The embedding of the quantum system within the classical system by using consistent equations of motion for the classical system with-proper conservation properties is discussed and applied to the one-dimensional collision of a classical particle with a quantum oscillator.
This paper presents a novel approach to quantum computing leveraging multi-scale modeling and simulation. The core claim is to develop a method capable of simultaneously simulating quantum systems at both microscopic and macroscopic levels, thereby enhancing the precision and efficiency of quantum computations. This is achieved through the construction of a multi-scale model integrating quantum mechanics and classical mechanics, coupled with advanced numerical simulation techniques. The proposed framework addresses the limitations of current quantum simulation methods by offering a more holistic representation of quantum systems. This allows for a deeper understanding of quantum phenomena and ultimately contributes to the advancement of quantum computing technologies. The key mathematical components utilized within this framework include the Schrödinger equation, perturbation theory, and various numerical integration schemes. Specifically, the time evolution of a quantum state is governed by the equation: (iħ ∂/∂t) |ψ(t)> = H |ψ(t)> where |ψ(t)> is the quantum state at time *t*, *ħ* is the reduced Planck constant, and *H* is the Hamiltonian operator. Perturbation theory, used to approximate the solution when the Hamiltonian is not solvable exactly, relies on the expansion: |ψperturbed>(t) = |ψunperturbed>(t) + ∑n=1∞ (-1)n+1 *An* |ψn>(t) where *An* are the perturbation coefficients. Numerical integration techniques, such as the Runge-Kutta method, are employed to approximate the time evolution operator. The accuracy of the simulation is critically dependent on the chosen time step, Δt, and the order of the integration method. A smaller Δt generally leads to higher accuracy but requires more computational resources. The overall approach seeks to bridge the gap between theoretical quantum mechanics and practical quantum computation, offering a viable path towards building more sophisticated and reliable quantum computers.
High Resolution Image Download MS PowerPoint Slide With a growing demand for time-domain simulations of correlated many-body systems, the development of efficient and stable integration schemes for the time-dependent Schrödinger equation is of keen interest in modern electronic structure theory. In this work, we present two approaches for the formation of the quantum propagator for time-dependent equation-of-motion coupled cluster theory based on the Chebyshev and Arnoldi expansions of the complex, nonhermitian matrix exponential, respectively. The proposed algorithms are compared with the short-iterative Lanczos method of Cooper et al. [ J. Phys. Chem. A 2021 125, 5438–5447], the fourth-order Runge–Kutta method, and exact dynamics for a set of small but challenging test problems. For each of the cases studied, both of the proposed integration schemes demonstrate superior accuracy and efficiency relative to the reference simulations.
Dissipation and decoherence, and the evolution from pure to mixed states in quantum physics, are handled through master equations for the density matrix. By embedding elements of this matrix in a higher-dimensional Liouville-Bloch equation, the methods of unitary integration are adapted to solve for the density matrix as a function of time, including the nonunitary effects of dissipation and decoherence. The input requires only solutions of classical, initial value time-dependent equations. Results are illustrated for a damped, driven two-level system.