The spin-boson (SB) model plays a central role in studies of dissipative quantum dynamics, due to bothits conceptual importance and relevance to a number of physical systems. Here, we provide rigorous bounds of the computational complexity of the SB model for the physically relevant case of a zero temperature ohmic bath. We start with the description of the bosonic bath via its Feynman-Vernon influence functional (IF), which is a tensor on the space of the trajectory of an impurity spin. By expanding the kernel of the IF via a sum of decaying exponentials, we obtain an analytical approximation of the continuous bath by a finite number of damped bosonic modes. We bound the error induced by restricting bosonic Hilbert spaces to a finite-dimensional subspace with small boson numbers, which yields an analytical form of a matrix-product state (MPS) representation of the IF. We show that the MPS bond dimension D scales polynomially in the error on physical observables as well as in the evolution time T, $D\propto T^4/{\epsilon }^2$. This bound indicates that the SB model can be efficiently simulated using a polynomial in time-computational resources.

中文翻译：

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时域张量网络近似非马尔可夫动力学的约束

自旋玻色子 (SB) 模型在耗散量子动力学研究中发挥着核心作用，因为它在概念上很重要，而且与许多物理系统相关。在这里，我们为零温欧姆浴的物理相关情况提供了 SB 模型计算复杂性的严格界限。我们首先通过其费曼-弗农影响泛函 (IF) 来描述玻色子浴，它是杂质自旋轨迹空间上的张量。通过衰减指数之和扩展 IF 的核，我们通过有限数量的阻尼玻色子模式获得了连续浴的解析近似。我们将通过将玻色子希尔伯特空间限制为具有小玻色子数的有限维子空间而引起的误差进行限制，这产生了 IF 的矩阵积状态 (MPS) 表示的解析形式。我们证明 MPS 键维数 D 在物理可观测量误差以及演化时间T中呈多项式缩放，$D\propto {\mathrm{时间}}^4/{\epsilon }^2$。该界限表明可以在时间计算资源中使用多项式有效地模拟 SB 模型。