We are interested in the simulation of open quantum systems governed by the Lindblad master equation in an infinite-dimensional Hilbert space. To simulate the solution of this equation, the standard approach involves two sequential approximations: first, we truncate the Hilbert space to derive a differential equation in a finite-dimensional subspace. Then, we use discrete time-step to obtain a numerical solution to the finite-dimensional evolution. In this paper, we establish bounds for these two approximations that can be explicitly computed to guarantee the accuracy of the numerical results. Through numerical examples, we demonstrate the efficiency of our method, empirically highlighting the tightness of the upper bound. While adaptive time-stepping is already a common practice in the time discretization of the Lindblad equation, we extend this approach by showing how to dynamically adjust the truncation of the Hilbert space. This enables fully adaptive simulations of the density matrix. For large-scale simulations, this approach can significantly reduce computational time and relieves users of the challenge of selecting an appropriate truncation.

翻译：我们关注无限维希尔伯特空间中由Lindblad主方程支配的开放量子系统的数值模拟。为模拟该方程的解，标准方法包含两个顺序逼近步骤：首先，我们截断希尔伯特空间以导出有限维子空间中的微分方程；随后，采用离散时间步长获得有限维演化的数值解。本文为这两种逼近建立了可显式计算误差界，从而保证数值结果的精度。通过数值算例，我们验证了所提方法的有效性，并经验性地证明了误差上界的紧致性。尽管自适应时间步长在Lindblad方程时间离散中已是常见实践，我们进一步拓展了该方法：通过展示如何动态调整希尔伯特空间的截断维度，实现了密度矩阵的完全自适应模拟。对于大规模模拟问题，该方法能显著减少计算时间，并免去用户手动选取合适截断维度的困难。