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

TL;DR

This paper establishes a theoretical bound on the computational complexity of simulating the spin-boson model, demonstrating that it can be efficiently approximated using tensor network methods with polynomial resource scaling.

Contribution

It provides the first rigorous bounds on the tensor network approximation of non-Markovian quantum dynamics in the spin-boson model at zero temperature.

Findings

Tensor network bond dimension scales polynomially with simulation time and desired accuracy.

Finite-dimensional approximation of the bosonic bath yields controlled error bounds.

The results imply efficient simulation of the spin-boson model is feasible.

Abstract

Spin-boson (SB) model plays a central role in studies of dissipative quantum dynamics, both due its 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 spin's trajectories. By expanding the kernel of the IF functional 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 $\epsilon$, as well as in the evolution time $T$, $D\propto T^4/\epsilon^2$. This bound indicates that the spin-boson model can be efficiently simulated using polynomial in time computational resources.