  Tensor networks states allow to find the low energy states of local lattice
Hamiltonians through variational optimization. Recently, a construction of such
states in the continuum was put forward, providing a first step towards the
goal of solving quantum field theories (QFTs) variationally. However, the
proposed manifold of continuous tensor network states (CTNSs) is difficult to
study in full generality, because the expectation values of local observables
cannot be computed analytically. In this paper, we study a tractable subclass
of CTNSs, the Gaussian CTNSs (GCTNSs), and benchmark them on simple quadratic
and quartic bosonic QFT Hamiltonians. We show that GCTNSs provide arbitrarily
accurate approximations to the ground states of quadratic Hamiltonians, and
decent estimates for quartic ones at weak coupling. Since they capture the
short distance behavior of the theories we consider exactly, GCTNSs even allow
to renormalize away simple divergences variationally. In the end, our study
makes it plausible that CTNSs are indeed a good manifold to approximate the low
energy states of QFTs.
