Provably Efficient Long-Time Exponential Decompositions of Non-Markovian Gaussian Baths

TL;DR

This paper provides rigorous bounds on the complexity of simulating non-Markovian Gaussian baths over long times, highlighting how spectral density features influence computational cost.

Contribution

It establishes time-uniform bounds on the number of exponentials needed for accurate bath correlation function representation, depending on spectral density singularities.

Findings

Number of exponentials is bounded independently of T for many spectral densities.

Polylogarithmic T-dependence appears only with strong spectral singularities.

Temperature effects are mild for bosonic baths and absent for fermionic baths.

Abstract

Gaussian baths are widely used to model non-Markovian environments, yet the cost of accurate simulation at long times remains poorly understood, especially when spectral densities exhibit nonanalytic behavior as in a range of realistic models. We rigorously bound the complexity of representing bath correlation functions on a time interval $[0,T]$ by sums of complex exponentials, as employed in recent variants of pseudomode and hierarchical equations of motion methods. These bounds make explicit the dependence on the maximal simulation time $T$, inverse temperature $\beta$, and the type and strength of singularities in an effective spectral density. For a broad class of spectral densities, the required number of exponentials is bounded independently of $T$, achieving time-uniform complexity. The $T$-dependence emerges only as polylogarithmic factors for spectral densities with strong singularities, such as step discontinuities and inverse power-law divergences. The temperature dependence is mild for bosonic environments and disappears entirely for fermionic environments. Thus, the true bottleneck for long-time simulation is not the simulation duration itself, but rather the presence of sharp nonanalytic features in the bath spectrum. Our results are instructive both for long-time simulation of non-Markovian open quantum systems, as well as for Markovian embeddings of classical generalized Langevin equations with memory kernels.