  In the study of open quantum systems, one of the most common ways to describe
environmental effects on the reduced dynamics is through the spectral density.
However, in many models this object cannot be computed from first principles
and needs to be inferred on phenomenological grounds or fitted to experimental
data. Consequently, some uncertainty regarding its form and parameters is
unavoidable; this in turn calls into question the accuracy of any theoretical
predictions based on a given spectral density. Here, we focus on the spin-boson
model as a prototypical open quantum system, and find two error bounds on
predicted expectation values in terms of the spectral density variation
considered, and state a sufficient condition for the strongest one to apply. We
further demonstrate an application of our result, by bounding the error brought
about by the approximations involved in the Hierarchical Equations of Motion
resolution method for spin-boson dynamics.
