  The hierarchical (multi-linear) rank of an order-$d$ tensor is key in
determining the cost of representing a tensor as a (tree) Tensor Network (TN).
In general, it is known that, for a fixed accuracy, a tensor with random
entries cannot be expected to be efficiently approximable without the curse of
dimensionality, i.e., a complexity growing exponentially with $d$. In this
work, we show that the ground state projection (GSP) of a class of unbounded
Hamiltonians can be approximately represented as an operator of low effective
dimensionality that is independent of the (high) dimension $d$ of the GSP. This
allows to approximate the GSP without the curse of dimensionality.
