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arXiv:1911.07374 · Physical review. E

Exponential integrators for stochastic Schrödinger equations

Jingze Li, Xiantao Li

Published Jan 28, 2020• 11 citations
arXivHTMLPDF

Abstract

We present a class of exponential integrators to compute solutions of the stochastic Schrödinger equations arising from the modeling of open quantum systems. To be able to implement the methods within the same framework as the deterministic counterpart, we express the solution using Kunita's representation. With appropriate truncations, the solution operator can be written as matrix exponentials, which can be efficiently implemented by the Krylov subspace projection. The accuracy is examined in terms of the strong convergence by comparing trajectories, and in terms of the weak convergence by comparing the density-matrix operators. We show that the local accuracy can be further improved by introducing third-order commutators in the exponential. The effectiveness of the proposed methods is tested using the example from Di Ventra et al. [J. Phys.: Condens. Matter 16, 8025 (2004)JCOMEL0953-898410.1088/0953-8984/16/45/024].

DOI
10.1103/physreve.101.013312

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We present a class of exponential integrators to compute solutions of the stochastic Schrödinger equations arising from the modeling of open quantum systems. To be able to implement the methods within the same framework as the deterministic counterpart, we express the solution using Kunita's representation. With appropriate truncations, the solution operator can be written as matrix exponentials, which can be efficiently implemented by the Krylov subspace projection. The accuracy is examined in terms of the strong convergence by comparing trajectories, and in terms of the weak convergence by comparing the density-matrix operators. We show that the local accuracy can be further improved by introducing third-order commutators in the exponential. The effectiveness of the proposed methods is tested using the example from Di Ventra et al. [J. Phys.: Condens. Matter 16, 8025 (2004)JCOMEL0953-898410.1088/0953-8984/16/45/024].
In this paper, we introduce a novel and general framework for the variational quantum simulation of Lindblad equations. Building on the close relationship between the unraveled Lindblad dynamics, stochastic Magnus integrators, and variational quantum simulation, we propose a high-order scheme for solving the quantum state diffusion equation using exponential integrators. This formulation facilitates the simulation of wavefunction trajectories within the established framework of variational quantum algorithms for time evolution. Our algorithm significantly enhances robustness in two key aspects: the stability of the simulation with large time steps, and the reduction in the number of quantum trajectories required to accurately simulate the Lindblad dynamics in terms of the ensemble average. We demonstrate the effectiveness of our algorithm through numerical examples in both classical and quantum implementations, including the transverse-field Ising model (TFIM) with damping, the Fenna-Matthews-Olson (FMO) complex, and the radical pair model (RPM). The simulation accuracy can be systematically improved, and the algorithm remains reliable even in highly oscillatory regimes. These methods are expected to be applicable to a broader class of open quantum systems beyond the specific models considered in this study.
The paper deals with the numerical solution of the nonlinear Itô stochastic differential equations (SDEs) appearing in the unravelling of quantum master equations. We first develop an exponential scheme of weak order 1 for general globally Lipschitz SDEs governed by Brownian motions. Then, we proceed to study the numerical integration of a class of locally Lipschitz SDEs. More precisely, we adapt the exponential scheme obtained in the first part of the work to the characteristics of certain finite-dimensional nonlinear stochastic Schrödinger equations. This yields a numerical method for the simulation of the mean value of quantum observables. We address the rate of convergence arising in this computation. Finally, an experiment with a representative quantum master equation illustrates the good performance of the new scheme.
By definition, an ensemble of quantum trajectories reproduces the density matrix of an open system after ensemble averaging. We present a general class of stochastic Schrodinger equations for quantum trajectories which unifies and extends existing methods. Various quantum jump and state-diffusion methods are recognized as opposite limiting cases. The inherent freedom of choice can be used to significantly reduce the calculation time in specific cases. Examples of this are methods with adaptive noise, for which a specific observable becomes noise free to first order. Linear stochastic equations can also be constructed, which allow analytical solutions. As an illustration, the decoherence of Schrodinger cat states of a cavity mode is discussed.
We present a universal framework for simulating NNN-dimensional linear Itô stochastic differential equations (SDEs) on quantum computers with additive or multiplicative noises. Building on a unitary dilation technique, we establish a rigorous mapping from the general linear SDEs dXt=A(t)Xt dt+∑j=1JBj(t)Xt dWtj dX_t = A(t) X_t\,dt + \sum_{j=1}^J B_j(t)X_t\,dW_t^j dXt​=A(t)Xt​dt+∑j=1J​Bj​(t)Xt​dWtj​ to stochastic Schrödinger equations (SSE) on a dilated Hilbert space. Crucially, this embedding is pathwise exact in that the classical solution is recovered as a projection of the dilated quantum state for each fixed noise realization. We demonstrate that the resulting SSEs are {naturally implementable} on digital quantum processors, where the stochastic Wiener increments are encoded directly by preparing the ancillary qubits. Exploiting this physical mapping, we develop two algorithmic strategies: (1) a trajectory-based approach that uses sequential weak measurements to realize efficient stochastic integrators, including a second-order scheme, and (2) an ensemble-based approach that maps moment evolution to a deterministic Lindblad quantum master equation, enabling simulation without Monte Carlo sampling. We provide error bounds based on a stochastic light-cone analysis and validate the framework with numerical experiments.
Abstract We address the numerical solution of the finite-dimensional non-linear stochastic Schrödinger equation, which is a locally Lipschitz stochastic differential equation modeling, for instance, quantum measurement processes. We study the rate of weak convergence of an exponential scheme that reproduces the norm of the desired solution by projecting onto the unit sphere. This justifies the use of the Talay-Tubaro extrapolation procedure in the numerical simulation of open quantum systems. In particular, we prove that an Euler-Exponential scheme converges with weak-order one, and we obtain the leading order term of its weak error expansion with respect to the step-size. Then, applying the Talay-Tubaro extrapolation procedure to the Euler-Exponential scheme under consideration we get a second-order method for computing the mean values of smooth functions of the solution of the non-linear stochastic Schrödinger equation. We also prove that the exponential scheme under study has order of strong convergence 1/2, which gives theoretical support to the use of the multilevel Monte Carlo method in simulating open quantum systems. We present a numerical experiment with a quantized electromagnetic field in interaction with a reservoir that illustrates the good performance of the weak second-order method, and the multilevel Monte Carlo method.
We study an explicit exponential scheme for the time discretisation of stochastic Schrödinger Equations Driven by additive or Multiplicative Itô Noise. The numerical scheme is shown to converge with strong order 1 if the noise is additive and with strong order 1/2 for multiplicative noise. In addition, if the noise is additive, we show that the exact solutions of the linear stochastic Schrödinger equations satisfy trace formulas for the expected mass, energy, and momentum (i. e., linear drifts in these quantities). Furthermore, we inspect the behaviour of the numerical solutions with respect to these trace formulas. Several numerical simulations are presented and confirm our theoretical results.