Focusing on the real-time dynamics of the reduced density matrix of the multidimensional Caldeira-Leggett model, several techniques are adopted in this paper to reduce the spatial and temporal dimensionality, combined into an efficient algorithm. From a spatial perspective, an equivalent formulation of the Dyson series is presented. With the aid of a low-rank approximation, the spatial dimensionality of open quantum system simulations is halved. From a temporal perspective, the frozen Gaussian approximation is used to approximate both the evolution operator and the interaction operator in the multidimensional Caldeira-Leggett model. This reduces the high-dimensional integrals to one- and two-dimensional integrals independent of the truncation level of the Dyson series. Through these techniques, we design an efficient algorithm whose validity is verified through several numerical experiments, including a two-dimensional double slit simulation.
Focusing on the real-time dynamics of the reduced density matrix of the multidimensional Caldeira-Leggett model, several techniques are adopted in this paper to reduce the spatial and temporal dimensionality, combined into an efficient algorithm. From a spatial perspective, an equivalent formulation of the Dyson series is presented. With the aid of a low-rank approximation, the spatial dimensionality of open quantum system simulations is halved. From a temporal perspective, the frozen Gaussian approximation is used to approximate both the evolution operator and the interaction operator in the multidimensional Caldeira-Leggett model. This reduces the high-dimensional integrals to one- and two-dimensional integrals independent of the truncation level of the Dyson series. Through these techniques, we design an efficient algorithm whose validity is verified through several numerical experiments, including a two-dimensional double slit simulation.
We propose an algorithm that combines the inchworm method and the frozen Gaussian approximation to simulate the Caldeira-Leggett model in which a quantum particle is coupled with thermal harmonic baths. In particular, we are interested in the real-time dynamics of the reduced density operator. In our algorithm, we use frozen Gaussian approximation to approximate the wave function as a wave packet in integral form. The desired reduced density operator is then written as a Dyson series, which is the series expression of path integrals in quantum mechanics of interacting systems. To compute the Dyson series, we further approximate each term in the series using Gaussian wave packets, and then employ the idea of the inchworm method to accelerate the convergence of the series. The inchworm method formulates the series as an integro-differential equation of “full propagators'', and rewrites the infinite series on the right-hand side using these full propagators, so that the number of terms in the sum can be significantly reduced, and faster convergence can be achieved. The performance of our algorithm is verified numerically by various experiments.
We present an efficient quantum algorithm for simulating the dynamics of Markovian open quantum systems. The performance of our algorithm is similar to the previous state-of-the-art quantum algorithm, i.e., it scales linearly in evolution time and poly-logarithmically in inverse precision. However, our algorithm is conceptually cleaner, and it only uses simple quantum primitives without compressed encoding. Our approach is based on a novel mathematical treatment of the evolution map, which involves a higher-order series expansion based on Duhamel's principle and approximating multiple integrals using scaled Gaussian quadrature. Our method easily generalizes to simulating quantum dynamics with time-dependent Lindbladians. Furthermore, our method of approximating multiple integrals using scaled Gaussian quadrature could potentially be used to produce a more efficient approximation of time-ordered integrals, and therefore can simplify existing quantum algorithms for simulating time-dependent Hamiltonians based on a truncated Dyson series.
We propose a general strategy to discretize the Dyson series without applying direct numerical quadrature to high-dimensional integrals and extend this framework to open quantum systems. The resulting discretization can also be interpreted as a Strang splitting combined with a Taylor expansion. Based on this formulation, we develop a deterministic iterative method for simulating system-bath dynamics. We propose two numerical schemes, which are first-order and second-order in the time step Δt, respectively. In the second-order scheme, we can safely omit most terms arising from the Strang splitting and Taylor expansion while maintaining second-order accuracy, leading to a substantial reduction in computational complexity. For the second-order method, we achieve a time complexity of O(M322KmaxKmax2) and a space complexity of O(M222KmaxKmax), where M denotes the number of system levels and Kmax the number of time steps within the memory length. Compared with existing methods, our approach requires substantially less memory and computational effort for multilevel systems (M ≥ 3). Numerical experiments are carried out to illustrate the validity and efficiency of our method.
A bstract We propose first-principle calculations of an open system based on the real-time path integral formalism treating the environment as well as the system of our interest together on a computer. The sign problem that occurs in applying Monte Carlo methods can be overcome in general by using the so-called Lefschetz thimble method, which has been developed over the past decade. Here we focus on the Caldeira-Leggett model, which is well known, in particular, as a model of quantum decoherence. In this case, the calculation simplifies drastically since the path integral becomes Gaussian for typical initial conditions. The relevant saddle point, which is unique and complex, can be determined by solving a linear equation with a huge but sparse coefficient matrix, and the integration over the Lefschetz thimble can be performed analytically. Thus we obtain, without assumptions or approximations, the reduced density matrix after a long-time evolution, tracing out a large number of harmonic oscillators in the environment. In particular, we confirm the dependence of the decoherence time on the coupling constant and the temperature that has been predicted from the master equation in a certain parameter regime.
We present a twofold contribution to the numerical simulation of Lindblad equations. First, an adaptive numerical approach to approximate Lindblad equations using low-rank dynamics is described: a deterministic low-rank approximation of the density operator is computed, and its rank is adjusted dynamically, using an on-the-fly estimator of the error committed when reducing the dimension. On the other hand, when the intrinsic dimension of the Lindblad equation is too high to allow for such a deterministic approximation, we combine classical ensemble averages of quantum Monte Carlo trajectories and a denoising technique. Specifically, a variance reduction method based on the consideration of a low-rank dynamics as a control variate is developed. Numerical tests for quantum collapse and revivals show the efficiency of each approach, along with the complementarity of the two approaches.
We develop LREI, short for Low-Rank Eigenmode Integration, a memory- and time-efficient numerical scheme for solving quantum Landau–Lifshitz (q-LL) and quantum Lan- dau–Lifshitz–Gilbert (q-LLG) equations, which govern spin dynamics in open quantum systems. Although the system size grows exponentially with the number of spins, our approach benefits from the low-rank structure of the density matrix and the sparsity of system Hamiltonians to avoid costly full matrix computations. By representing density matrices in terms of their low-rank factors and using Krylov subspace techniques for partial eigendecompositions, we reduce the per-step complexity of Runge–Kutta and Adams–Bashforth schemes from O(N 3) to O(r2N ), where N = 2n is the Hilbert space dimension for n spins and r ? N is the effective rank of the density matrix. Likewise, the memory footprint is reduced from O(N 2) to O(rN ), since no full N × N matrices are ever formed. Among several technical improvements we applied, one key idea was to handle the computation of the action of the invariant subspace of the density matrix associated with its zero eigenvalues. This was accomplished by applying Householder reflectors constructed for the dominant eigenspace, thereby enabling the entire solution process to proceed without ever forming any large matrices. As an example, we can now evolve a time step of a twenty-spin system—corresponding to a density matrix size exceeding one million—in just a few seconds on a standard laptop. In addition, both classes of Runge-Kutta and Adams-Bashforth techniques are reformulated to preserve the physical properties of the density matrix throughout the time evolution. The new low-rank algorithm makes it possible to simulate much larger spin systems that were previously computationally infeasible. This, in turn, provides a powerful tool for comparing the q-LL and q-LLG dynamics, assessing the validity of each model, and exploring how quantum features such as correlations and entanglement evolve across different regimes of system size and damping.