  Constructing optimal thermodynamic processes in quantum systems relies on
managing the balance between the average excess work and its stochastic
fluctuations. Recently it has been shown that two different quantum
generalisations of thermodynamic length can be utilised to determine protocols
with either minimal excess work or minimal work variance. These lengths measure
the distance between points on a manifold of control parameters, and optimal
protocols are achieved by following the relevant geodesic paths given some
fixed boundary conditions. Here we explore this problem in the context of
Gaussian quantum states that are weakly coupled to an environment and derive
general expressions for these two forms of thermodynamic length. We then use
this to compute optimal thermodynamic protocols for various examples of
externally driven Gaussian systems with multiple control parameters.
