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Spin-boson (SB) model plays a central role in studies of dissipative quantumdynamics, both due its conceptual importance and relevance to a number ofphysical systems. Here we provide rigorous bounds of the computationalcomplexity of the SB model for the physically relevant case of a zerotemperature Ohmic bath. We start with the description of the bosonic bath viaits Feynman-Vernon influence functional (IF), which is a tensor on the space ofspin's trajectories. By expanding the kernel of the IF functional via a sum ofdecaying exponentials, we obtain an analytical approximation of the continuousbath by a finite number of damped bosonic modes. We bound the error induced byrestricting bosonic Hilbert spaces to a finite-dimensional subspace with smallboson numbers, which yields an analytical form of a matrix-product state (MPS)representation of the IF. We show that the MPS bond dimension $D$ scalespolynomially in the error on physical observables $\epsilon$, as well as in theevolution time $T$, $D\propto T^4/\epsilon^2$. This bound indicates that thespin-boson model can be efficiently simulated using polynomial in timecomputational resources.

A bound on approximating non-Markovian dynamics by tensor networks in the time domain